2.1 Coding theory

We briefly recall some definitions from coding theory. A discrete metric space is a pair $(\mathcal {X},d),$ where $\mathcal {X}$ is a finite set and $d:\mathcal {X} \times \mathcal {X} \to {\mathbb {R}}_{\geq 0}$ is a function such that:

• • for all $\textbf {x},\textbf {y} \in \mathcal {X,}$ we have $d(\textbf {x},\textbf {y})=0 \Leftrightarrow \textbf {x} = \textbf {y}$ ;

• • for all $\textbf {x},\textbf {y} \in \mathcal {X,}$ we have $d(\textbf {x},\textbf {y})=d(\textbf {y},\textbf {x})$ ;

• • for all $\textbf {x},\textbf {y},\textbf {z} \in \mathcal {X,}$ we have $d(\textbf {x},\textbf {z}) \leq d(\textbf {x},\textbf {y}) + d(\textbf {y},\textbf {z})$ , which is the triangle inequality.

The classic example of a discrete metric space in coding theory is the set ${\mathbb {F}}_q^n$ with the Hamming metric $d_{\text {H}}$ , which is defined as $d_{\text {H}}(\textbf {x},\textbf {y}) := |\{1 \leq i \leq n: x_i \neq y_i\}|$ for $\textbf {x}=(x_1,\ldots , x_n), \textbf {y} =(y_1, \ldots , y_n) \in {\mathbb {F}}_q^n$ .

A code is a subset $\mathcal {C} \subseteq \mathcal {X}$ with $|\mathcal {C}| \geq 2$ . The elements of a code $\mathcal {C}$ are called code words. The minimum distance of a code $\mathcal {C} \subseteq \mathcal {X}$ is defined as

$$\begin{align*}d(\mathcal{C}) := \min \{d(\textbf{x},\textbf{y}) \mid \textbf{x},\textbf{y} \in \mathcal{C}, \, \textbf{x} \neq \textbf{y}\}. \end{align*}$$

The main problem of classical coding theory is understanding how large a code of certain minimum distance can be. In this regard, the largest cardinality of a code $\mathcal {C} \subseteq {\mathbb {F}}_q^n$ of minimum distance d is denoted as $A_q(n,d)$ . For the Hamming metric, several upper bounds exist for this quantity $A_q^{\text {H}}(n,d)$ , e.g., the Singleton bound [Reference Huffman and Pless36, Theorem 2.4.1], the Hamming bound (or sphere-packing bound) [Reference Huffman and Pless36, Theorem 1.12.1], and the Plotkin bound [Reference Huffman and Pless36, Theorem 2.2.1]. On the other hand, code constructions can give lower bounds for $A_q^{\text {H}}(n,d)$ (see [Reference Huffman and Pless36, Reference MacWilliams and Sloane43] among others).

While the problem of computing the maximum cardinality of a code with given minimum distance has been extensively studied for the Hamming metric, the question is less understood for other metrics. The Lee metric, the rank metric, and the sum-rank metric, among others, are examples of metrics that are also often used in coding theory. Sum-rank-metric codes, for instance, have been used for multi-shot network coding [Reference Martínez-Peñas and Kschischang44] and space-time coding [Reference Shehadeh and Kschischang50]. More discrete metric spaces follow in the remainder of this article.

2.2 Graph theory

Next, we recall some notions of graph theory, with a special focus on the graph properties that are used in the rest of this article.

A graph automorphism of a graph $G=(V,E)$ is a permutation $\sigma $ of the vertex set V such that $(x,y) \in E$ if and only if $(\sigma (x), \sigma (y)) \in E$ . A graph is vertex-transitive if for any two vertices $x,y,$ there exists a graph automorphism $\sigma $ such that $\sigma (x)=y$ . Note that vertex-transitive graphs are regular.

A graph is a Cayley graph over a group G with connecting set S if the vertices of the graph are the elements of G, and two vertices $x,y$ are adjacent if and only if there is an element $s \in S$ such that $x+s=y$ . In this work, we assume that the connecting set S does not contain the identity element of G and that S is closed under inverses. This assumption implies that the corresponding Cayley graph is undirected and has no self-loops. Note that Cayley graphs are vertex-transitive.

Note that vertex-transitive graphs are necessarily walk-regular.

For any two vertices x and y at distance i from each other, let $p_{j,h}^i(x,y)$ denote the number of vertices at distance j from x and at distance h from y.

In particular, a graph is k-partially distance-regular if for any integer $i \leq k,$ the values $c_i(x,y):=p_{1,i-1}^i(x,y)$ , $a_{i-1}(x,y):=p_{1,i-1}^{i-1}(x,y)$ , and $b_{i-2} := p_{1,i-1}^{i-2}(x,y)$ do not depend on the choice of x and y. For distance-regular graphs, these values are captured in the intersection array $(b_0, b_1, \ldots , b_{D-1}; c_1, \ldots , c_D)$ , where D is the diameter of the graph. Since distance-regular graphs are $\delta $ -regular for some $\delta \in \mathbb {N}_0$ , the following relations hold: $a_i+b_i+c_i=\delta $ for $0 \leq i \leq D$ , $b_0=\delta $ , $a_0=c_0=0$ . Note that k-partially distance-regular graphs are also k-partially walk-regular.

Recall the Cartesian product of two graphs G and H, denoted as $G \Box H$ , which is the graph with vertex set equal to the Cartesian product of the vertex sets of G and H, where two vertices $(g_1,h_1)$ and $(g_2,h_2)$ are adjacent if $g_1 \sim g_2$ and $h_1=h_2$ , or $g_1=g_2$ and $h_1 \sim h_2$ . Here $\sim $ denotes adjacency of the vertices in the graph. The Cartesian product of two graphs can be inductively extended to a Cartesian product of finitely many graphs. It is well-known that if G and H are graphs with respective eigenvalues $\lambda _i, i \in I$ and $\mu _j, j \in J$ , then the eigenvalues of $G \Box H$ are $\lambda _i + \mu _j$ for $i \in I, j \in J$ (see, for instance, [Reference Cvetković, Doob and Sachs20]). This can be inductively extended to the Cartesian product of finitely many graphs.

Alternatively, we can consider the k-th power graph $G^k$ of a graph $G=(V,E)$ , which is the graph with vertex set $V,$ where two vertices $x,y \in V$ are adjacent if $d_G(x,y) \leq k$ . Here, $d_G(x,y)$ denotes the geodesic distance between vertices x and y in the graph G. The k-independence number of G equals the ( $1$ -)independence number of $G^k$ , which is the size of the largest independent set in $G^k$ . Despite this, even the simplest algebraic or combinatorial parameters of the power graph $G^k$ cannot be easily deduced from the corresponding parameters of the graph G, e.g., neither the spectrum [Reference Abiad, Coutinho, Fiol, Nogueira and Zeijlemaker3, Reference Das and Guo21], nor the average degree [Reference DeVos, McDonald and Scheide26], nor the rainbow connection number [Reference Basavaraju, Chandran, Rajendraprasad and Ramaswamy10] of $G^k$ can be derived in general directly from those of the original graph G. In this regard, several eigenvalue bounds on $\alpha _k(G)$ that only depend on the spectrum of G have been proposed in the literature. Another upper bound on the independence number, and after extension the k-independence number, of a graph is the Lovász theta number [Reference Lovász42], although this bound requires the graph adjacency matrix as input. The Lovász theta number can be used as an upper bound on the k-independence number of a graph G by computing it for the k-th power graph $G^k$ .

The k-independence number of a graph and these eigenvalue bounds are the bases on which the Eigenvalue Method is built, as we see in the next section.

3.1 Eigenvalue bounds

In this section, we give the eigenvalue bounds that are used in the Eigenvalue Method. First, the Inertia-type bound and its MILP implementation are given. Then, the Ratio-type bound and its LP implementation are stated.

Define $\mathbb {R}_k[x]$ as the set of all polynomials in the variable x with real coefficients and degree at most k. The Inertia-type bound is an upper bound on the k-independence number of a graph.

Theorem 3.2 (Inertia-type bound, [Reference Abiad, Coutinho and Fiol2, Theorem 3.1])

Let G be a graph with n vertices, adjacency eigenvalues $\lambda _1 \geq \cdots \geq \lambda _n$ , and adjacency matrix A. Let $p \in \mathbb {R}_k[x]$ with corresponding parameters $W(p):= \max _{u \in V(G)} \{(p(A))_{uu}\}$ , $w(p):= \min _{u \in V(G)} \{(p(A))_{uu}\}$ . Then, the k-independence number $\alpha _k$ of G satisfies

$$\begin{align*}\alpha_k \leq \min \left\{|\{i: p(\lambda_i) \geq w(p)\}|, \, |\{i: p(\lambda_i) \leq W(p)\}| \right\}. \end{align*}$$

Note that for $k=1,$ the Inertia-type bound reduces to the well-known inertia bound by Cvetković [Reference Cvetković18].

In [Reference Abiad, Coutinho, Fiol, Nogueira and Zeijlemaker3] an MILP has been proposed that finds the optimal polynomial for the Inertia-type bound, which is the polynomial that minimizes the upper bound on the k-independence number. This MILP subsequently finds this minimized upper bound on the k-independence number. Let G be a graph with n vertices, distinct adjacency eigenvalues $\theta _0> \cdots > \theta _r$ with respective multiplicities $m_0, \ldots , m_r$ , and adjacency matrix A. Let $p(x) := a_kx^k + \cdots + a_0$ , $\textbf {b}=(b_0, \ldots , b_r) \in \{0,1\}^{r+1}$ , and $\textbf {m} = (m_0, \ldots , m_r)$ . Then, the following MILP with variables $a_0, \ldots , a_k$ and $b_0, \ldots , b_r$ finds the optimal polynomial for Theorem 3.2:

Here, M is some fixed large number and $\epsilon>0$ is small. This MILP has to run for every $u \in V(G)$ and the best objective value is then the minimum of all separate objective values. This lowest objective value is exactly the best upper bound for the k-independence number that can be obtained from the Inertia-type bound.

In [Reference Abiad, Coutinho, Fiol, Nogueira and Zeijlemaker3] it is discussed that if the graph G is k-partially walk-regular, then MILP (3.1) only has to run for one $u \in V(G)$ . In this case, using the same notation as above, MILP (3.1) simplifies to the following MILP:

For k-partially walk-regular graphs G, the objective value of MILP (3.2) is exactly the best upper bound for the k-independence number that can be obtained from the Inertia-type bound.

The Ratio-type bound is another upper bound on the k-independence number, but specifically for regular graphs.

Theorem 3.3 (Ratio-type bound, [Reference Abiad, Coutinho and Fiol2, Theorem 3.2])

Let G be a regular graph with n vertices, adjacency eigenvalues $\lambda _1 \geq \cdots \geq \lambda _n$ , and adjacency matrix A. Let $p \in \mathbb {R}_k[x]$ with corresponding parameters $W(p):= \max _{u \in V(G)} \{(p(A))_{uu}\}$ , $\lambda (p):= \min _{i \in [2,n]} \{p(\lambda _i)\}$ . Assume that $p(\lambda _1)> \lambda (p)$ . Then, the k-independence number $\alpha _k$ of G satisfies

$$\begin{align*}\alpha_k \leq n \frac{W(p)-\lambda(p)}{p(\lambda_1) - \lambda(p)}. \end{align*}$$

For $k=1,$ the Ratio-type bound can be reduced to the well-known ratio bound by Hoffman (unpublished, see, e.g., [Reference Haemers34, Theorem 3.2]).

For $k=2,3,$ there are closed-form expressions for the Ratio-type bound that no longer depend on the choice of $p \in \mathbb {R}_k[x]$ and that are optimal in the sense that no better bound can be obtained via Theorem 3.3.

Theorem 3.4 [Reference Abiad, Coutinho and Fiol2, Corollary 3.3]

Let G be a regular graph with n vertices and distinct adjacency eigenvalues $\theta _0> \theta _1 > \cdots > \theta _r$ with $r \geq 2$ . Let $\theta _i$ be the largest eigenvalue such that $\theta _i \leq -1$ . Then, the 2-independence number $\alpha _2$ of G satisfies

$$\begin{align*}\alpha_2 \leq n \frac{\theta_0 + \theta_i \theta_{i-1}}{(\theta_0 - \theta_i)(\theta_0 - \theta_{i-1})}. \end{align*}$$

Moreover, this is the best possible bound that can be obtained by choosing a polynomial via Theorem 3.3.

Theorem 3.5 [Reference Kavi and Newman38, Theorem 11]

Let G be a regular graph with n vertices, distinct adjacency eigenvalues $\theta _0> \theta _1 > \cdots > \theta _r$ with $r \geq 3$ , and adjacency matrix A. Let $\theta _s$ be the smallest eigenvalue such that $\theta _s \geq - \frac {\theta _0^2 + \theta _0 \theta _r - \Delta }{\theta _0 (\theta _r+1)}$ , where $\Delta = \max _{u \in V(G)} \{(A^3)_{uu}\}$ . Then, the 3-independence number $\alpha _3$ of G satisfies

$$\begin{align*}\alpha_3 \leq n \frac{\Delta - \theta_0 (\theta_s + \theta_{s+1} + \theta_r) - \theta_s \theta_{s+1} \theta_r}{(\theta_0 - \theta_s)(\theta_0 - \theta_{s+1})(\theta_0 - \theta_r)}. \end{align*}$$

Moreover, this is the best possible bound that can be obtained by choosing a polynomial via Theorem 3.3.

In [Reference Fiol30] an LP has been proposed that finds the optimal polynomial for the Ratio-type bound, which is the polynomial that minimizes the upper bound on the k-independence number, in case, the graph G is k-partially walk-regular. The objective value of this LP, which uses the so-called minor polynomials, subsequently equals this minimized upper bound. Let G be a k-partially walk-regular graph with distinct adjacency eigenvalues $\theta _0> \theta _1 > \cdots > \theta _r$ with respective multiplicities $m_0, m_1, \ldots , m_r$ . The k-minor polynomial is the polynomial $f_k \in \mathbb {R}_k[x]$ that minimizes $\sum _{i=0}^r m_i f(\theta _i)$ . Define the polynomial $f_k$ as $f_k(\theta _0) := x_0 = 1$ and $f_k(\theta _i) := x_i$ for $i=1, \ldots , r$ , where $(x_1, \ldots , x_r)$ is a solution of the following LP:

Here, $f[\theta _0, \ldots , \theta _s]$ denotes the s-th divided difference of Newton interpolation, recursively defined by

$$\begin{align*}f[\theta_i, \ldots, \theta_j] := \frac{f[\theta_{i+1}, \ldots, \theta_j] - f[\theta_i, \ldots, \theta_{j-1}]}{\theta_j - \theta_i}, \end{align*}$$

where $j>i$ , starting with $f[\theta _i] = f(\theta _i) = x_i$ for $i=0,1,\ldots , r$ . The best upper bound for the k-independence number of k-partially walk-regular graphs that can be obtained via the Ratio-type bound is exactly the objective value of LP (3.3), which follows from [Reference Fiol30, Theorem 4.1].

Note that the Lovász theta number is at least as good as the Ratio-type bound, but the Ratio-type bound can be computed exactly and more efficiently since it only requires the graph spectrum, while the Lovász theta number is an approximation and requires the graph adjacency matrix. The relation between the performance of the Lovász theta number and the Inertia-type bound is not known.

4.1 City block metric

The city block metric, also called the $L^1$ -metric or the Manhattan metric, was used already in the 18th century. Its first appearance in a coding-theoretical context is in [Reference Ulrich52], although it is not properly defined as a metric yet. The city block metric can be viewed as an extension of the Lee metric, which is widely used in coding theory, to $\mathbb {Z}^n$ instead of ${\mathbb {F}}_q^n$ .

Note that for $m=2,$ the city block metric coincides with the Hamming metric. Therefore, we assume $m \geq 3$ .

Now, we consider the discrete metric space $([\![m-1]\!]^n, d_{\text {cb}})$ and apply the Eigenvalue Method. Define the city block distance graph $G_{\text {cb}}([\![m-1]\!]^n)$ as the graph with vertex set $[\![m-1]\!]^n$ , where vertices $\textbf {x},\textbf {y} \in [\![m-1]\!]^n$ are adjacent if $d_{\text {cb}}(\textbf {x},\textbf {y})=1$ . First, we verify that condition (C1) holds for this graph.

Since condition (C1) holds, the Eigenvalue Method is applicable to the discrete metric space $([\![m-1]\!]^n, d_{\text {cb}})$ . Next, we check the desired properties (P1)–(P3).

Remark 4.2 shows that properties (P1) and (P2) are not satisfied, while (P3) is. This implies that only the Inertia-type bound is applicable to the graph $G_{\text {cb}}([\![m-1]\!]^n)$ and not the Ratio-type bound as it requires regularity of the graph. In order to apply the former bound, we first determine the adjacency eigenvalues of the graph. These eigenvalues follow directly from the next result.

Proof of Lemma 4.3

Fix m. We prove the result by induction on n.

For $n=1,$ the graph $G_{\text {cb}}([\![m-1]\!]^n)$ has m vertices indexed as $0,1, \ldots , m-1$ and edge set $\{(i,i+1): i=0, \ldots , m-2\}$ . So $G_{\text {cb}}([\![m-1]\!]^n)$ indeed equals the path graph on m vertices (after renumbering of the vertices).

Now suppose that the city block distance graph for the discrete metric space $([\![m-1]\!]^n, d_{\text {cb}})$ equals the Cartesian product of n path graphs on m vertices:

$$\begin{align*}\underbrace{P_m \Box \cdots \Box P_m}_{n \text{ times}} =: G, \end{align*}$$

where $P_m$ denotes the path graph on m vertices. Consider the city block distance graph $G_{\text {cb}}([\![m-1]\!]^{n+1})$ , so for $n+1$ . The vertex set of $G \Box P_m$ equals the vertex set of $G_{\text {cb}}([\![m-1]\!]^{n+1})$ (given the right naming of the vertices of $P_m$ , namely, 0 through $m-1$ ). Let $\textbf {x}=(x_1,\ldots , x_{n+1}),\textbf {y}=(y_1,\ldots , y_{n+1}) \in [\![m-1]\!]^{n+1}$ be two adjacent vertices in $G \Box P_m$ . Then,

• • $(x_1, \ldots , x_n) = (y_1, \ldots , y_n)$ and $x_{n+1} \sim y_{n+1}$ so $d_{\text {cb}}((x_1, \ldots , x_n), (y_1, \ldots , x_n) = 0$ and $d_{\text {cb}}(x_{n+1}, y_{n+1}) = 1$ , or

• • $(x_1, \ldots , x_n) \sim (y_1, \ldots , y_n)$ and $x_{n+1} = y_{n+1}$ so $d_{\text {cb}}((x_1, \ldots , x_n), (y_1, \ldots , x_n) = 1$ and $d_{\text {cb}}(x_{n+1}, y_{n+1}) = 0$ .

In both cases $d_{\text {cb}}(\textbf {x},\textbf {y}) = 1$ , so $\textbf {x}$ and $\textbf {y}$ are adjacent in $G_{\text {cb}}([\![m-1]\!]^{n+1})$ . Moreover, these are the only two options when $\textbf {x}, \textbf {y} \in [\![m-1]\!]^{n+1}$ are adjacent in $G_{\text {cb}}([\![m-1]\!]^{n+1})$ . Hence, the graph $G_{\text {cb}}([\![m-1]\!]^{n+1})$ is equal to the Cartesian product of graphs G and $P_m$ . By induction, $G_{\text {cb}}([\![m-1]\!]^{n})$ equals the Cartesian product of n path graphs on m vertices.

Now, we are ready to derive the eigenvalue of our graph of interest.

Lemma 4.4 The adjacency eigenvalues of $G_{\text {cb}}([\![m-1]\!]^n)$ are

$$\begin{align*}\lambda_{\textbf{k}} = \sum_{j=1}^n 2\cos \left(\frac{k_j \pi}{m+1} \right) \end{align*}$$

for every tuple $\textbf {k} = (k_1, \ldots , k_n) \in [m]^n$ .

This expression for the eigenvalues of the city block distance graph can now be used in the Inertia-type bound to obtain new bounds on the maximum cardinality of codes in the city block metric. We then compare the obtained bounds to state-of-the-art bounds: the Plotkin-type bound and the Hamming-type bound.

Theorem 4.5 (Plotkin-type bound, [Reference Berlekamp11, Theorem 13.49])

Let $\mathcal {C} \subseteq [\![m-1]\!]^n$ be a code of minimum city block distance d. If $d> \frac {n(m-1)}{2}$ , then

$$\begin{align*}|\mathcal{C}| \leq \frac{2d}{2d-n(m-1)}. \end{align*}$$

Theorem 4.7 (Hamming-type bound, [Reference García-Claro and Gutiérrez33, Theorem 3.1])

Let $\mathcal {C} \subseteq [\![m-1]\!]^n$ be a code of minimum city block distance d. Define $t:= \lfloor \tfrac {d-1}{2} \rfloor $ . Then,

$$\begin{align*}|\mathcal{C} | \leq \frac{m^n}{\eta_t \left( [\![m-1]\!]^n \right) }, \end{align*}$$

where $\eta _t([\![m-1]\!]^n) := \min \left \{ |B_t(\textbf {x})|: \textbf {x} \in [\![m-1]\!]^n \right \}$ and $B_t(\textbf {x}) := \{\textbf {y} \in [\![m-1]\!]^n: d_{\text {cb}}(\textbf {x},\textbf {y}) \leq t\}$ .

In [Reference García-Claro and Gutiérrez33, Algorithm 1] an algorithm is described that computes the value of $\eta _t([\![m-1]\!]^n)$ given specific values of t and $\textbf {x}$ . We use this algorithm to compute the Hamming-type upper bound in specific instances.

Next, we compare the bound obtained using the Inertia-type bound with the Plotkin-type bound from Theorem 4.5 and the Hamming-type bound from Theorem 4.7. We split the discussion in two cases.

Case $k=1$ : For $k=1,$ the Inertia-type bound reduces to the well-known inertia bound, which is independent of the choice of the polynomial $p \in {\mathbb {R}}_k[x]$ . Therefore, the case $k=1$ is considered first. In this case $d=k+1=2$ , so for the Hamming-type bound, we get $t:=\lfloor \tfrac {d-1}{2} \rfloor =0$ and $\eta _t([\![m-1]\!]^n) = 1$ since any ball around a code word of radius 0 contains only the code word itself. The Hamming-type bound then becomes $|\mathcal {C}| \leq m^n$ , which is a trivial upper bound, so the inertia bound certainly performs better than the Hamming-type bound. Since $d=2$ , the condition of the Plotkin-type bound, $d>\tfrac {n(m-1)}{2}$ , together with the constraint $m \geq 3$ , gives only two instances where the Plotkin-type bound applies: $n=1, m=3$ and $n=1,m=4$ . For $n=1, m=3,$ both the inertia bound and the Plotkin-type bound give an upper bound of 2. For $n=1,m=4,$ the inertia bound gives 2, while the Plotkin-type bound gives 4. So in this instance, the inertia bound gives an improved upper bound. Note that in both instances, the inertia bound turns out to be sharp. All in all, the inertia bound, which is the Inertia-type bound in the case $k=1$ , performs no worse than the Plotkin-type bound and the Hamming-type bound.

Case $k \geq 2$ : Next, we consider the case $k \geq 2$ . Here, we resort to the proposed MILP for computing the value of the Inertia-type bound in specific instances. Since the city block distance graph is not walk-regular, only the slower MILP (3.1) is applicable. This MILP requires the construction of the graph $G_{\text {cb}}([\![m-1]\!]^n)$ to use the adjacency matrix of this graph. In this case, we also compute the Lovász theta number of the graph.

Now, we compare the upper bounds from the Inertia-type bound with the Plotkin-type bound, the Hamming-type bound, and the Lovász theta number for the following instances:

$$\begin{align*}n =1,2,3, \quad m =3,4, \quad k=1,\ldots, n(m-1)-1, \end{align*}$$

$$\begin{align*}\text{and } n=1,2, \quad m=5,6, \quad k=1, \ldots n(m-1)-1, \end{align*}$$

$$\begin{align*}\text{and } n=3, \quad m=5, \quad k=1,\ldots,7. \end{align*}$$

Note that we are also considering some instances where $k=1$ since we have only seen two of those thus far. The results can be seen in Table 2. The column “Inertia-type” gives the output of the Inertia-type bound for the given graph instance. Similarly, the column “ $\vartheta (G^k)$ ” contains the value of the Lovász theta number, the column “Plotkin-type” contains the value of the Plotkin-type upper bound, and the column “Hamming-type” contains the value of the Hamming-type bound. The column “ $\alpha _k$ ” contains the value of the k-independence number of the graph for that instance. Only the instances where the Inertia-type bound performed no worse than the Plotkin-type bound and the Hamming-type bound are present in the table. An upper bound in the column “Inertia-type” is marked in bold when it is less than both the Plotkin-type upper bound (if applicable) and the Hamming-type upper bound.

Table 2: Results of the Inertia-type bound for the city block metric, compared to the Plotkin-type bound, the Hamming-type bound, the Lovász theta number $\vartheta (G^k)$ , and the actual k-independence number $\alpha _k$ . Improvements of the Inertia-type bound compared to the Plotkin-type bound and the Hamming-type bound are marked in bold.

We see that there are several instances where the Inertia-type bound performs better than both the Plotkin-type bound and the Hamming-type bound. In all of these instances, the Inertia-type bound also performs as good as the Lovász theta number and is sharp. Moreover, there are many other instances where the Inertia-type bound is also sharp.

4.2 Projective metric

The projective metric, introduced in [Reference Gabidulin and Simonis32] and recently investigated in [Reference Riccardi and Sauerbier Couvée46], is a general metric that depends on a specific choice of set. Many well-known metrics are instances of this metric for the right choice of set, like the Hamming metric and the sum-rank metric.

Definition 4.2 Let $\mathcal {F}=\{F_1, \ldots , F_m\}$ be a set of one-dimensional subspaces of ${\mathbb {F}}_q^n$ such that $\text {span} \left ( \bigcup _{i=1}^m F_i \right ) = {\mathbb {F}}_q^n$ . The projective $\mathcal {F}$ -weight of $\textbf {x} \in {\mathbb {F}}_q^n$ is defined as

$$\begin{align*}w_{\mathcal{F}}(\textbf{x}) := \min \left\{|I|: \textbf{x} \in \text{span} \left( \bigcup_{i \in I} F_i \right)\right\}. \end{align*}$$

The projective $\mathcal {F}$ -distance between $\textbf {x},\textbf {y} \in {\mathbb {F}}_q^n$ is defined as $d_{\mathcal {F}}(\textbf {x},\textbf {y}) := w_{\mathcal {F}}(\textbf {x}-\textbf {y})$ .

From here on, let $\mathcal {F}$ be such a set as in Definition 4.2. We consider the discrete metric space $({\mathbb {F}}_q^n, d_{\mathcal {F}})$ and apply the Eigenvalue Method to it. Define the projective $\mathcal {F}$ -distance graph $G_{\mathcal {F}}({\mathbb {F}}_q^n)$ as the graph with vertex set ${\mathbb {F}}_q^n$ , where vertices $\textbf {x},\textbf {y} \in {\mathbb {F}}_q^n$ are adjacent if $d_{\mathcal {F}}(\textbf {x},\textbf {y})=1$ . We need to verify first if condition (C1) is satisfied.

Proof Let $\textbf {x},\textbf {y} \in {\mathbb {F}}_q^n$ with $d_{\mathcal {F}}(\textbf {x},\textbf {y})=d$ . Then, by definition, there is a subset $\{i_1, \ldots ,i_d\} \subseteq [m]$ of size d such that

$$\begin{align*}\textbf{x}-\textbf{y} \in \text{span} \left( \bigcup_{j=1}^d F_{i_j} \right). \end{align*}$$

This means that for all $j \in \{1,\ldots , d\},$ there exists an $\textbf {f}_{i_j} \in F_{i_j}$ such that $\textbf {x}-\textbf {y} = \sum _{j=1}^d \textbf {f}_{i_j}$ . Note that $(\textbf {x}, \textbf {x}-\textbf {f}_{i_1}, \textbf {x}-\textbf {f}_{i_1}-\textbf {f}_{i_2}, \ldots , \textbf {x}-\sum _{j=1}^d \textbf {f}_{i_j} = \textbf {y})$ is a path in $G_{\mathcal {F}}({\mathbb {F}}_q^n)$ from $\textbf {x}$ to $\textbf {y}$ of length d. So the geodesic distance between $\textbf {x}$ and $\textbf {y}$ is at most d. By minimality of the cardinality of $\{i_1, \ldots , i_d\},$ the geodesic distance cannot be less than d. Hence, the geodesic distance in $G_{\mathcal {F}}({\mathbb {F}}_q^n)$ equals the projective $\mathcal {F}$ -distance.

Since condition (C1) is satisfied, the Eigenvalue Method is applicable. The next results show that the projective $\mathcal {F}$ -distance graph has desired properties (P1) and (P2).

The last property to check is 3. However, distance-regularity of the graph $G_{\mathcal {F}}({\mathbb {F}}_q^n)$ depends on the choice of set $\mathcal {F}$ . The set $\mathcal {F}_{\text {H}} = \{F_1, \ldots , F_n\}$ with $F_i = \text {span}(\textbf {e}_i)$ for $i=1, \ldots , n$ , for instance, results in the Hamming distance, and the corresponding graph $G_{\mathcal {F}_{\text {H}}}({\mathbb {F}}_q^n)$ is the Hamming graph, which is known to be distance-regular (see, e.g., [Reference Brouwer, Cohen and Neumaier15]). On the other hand, the sum-rank metric, which we elaborate on below, results in a graph that is, in most instances, not distance-regular [Reference Abiad, Khramova and Ravagnani5, Proposition 12].

The sum-rank metric is an example of a projective metric.

Definition 4.3 Let t be a positive integer and let $\textbf {n}=(n_1,\ldots ,n_t)$ , $\textbf {m} = (m_1,\ldots ,m_t)$ be ordered tuples of positive integers with $m_1 \geq m_2 \geq \cdots \geq m_t$ , and $m_i\geq n_i$ for all $i\in [t]$ . The sum-rank-metric space is an $\mathbb {F}_q$ -linear vector space $\mathbb {F}_q^{\mathbf {n}\times \mathbf {m}}$ defined as follows:

$$\begin{align*}\mathbb{F}_q^{\mathbf{n}\times \mathbf{m}}:= \mathbb{F}_q^{n_1\times m_1}\times \cdots \times \mathbb{F}_q^{n_t\times m_t}. \end{align*}$$

The sum-rank of an element $X=(X_1,\ldots , X_t)\in \mathbb {F}_q^{\mathbf {n}\times \mathbf {m}}$ is $\mathrm {srk}(X) := \sum _{i=1}^t \operatorname {rk}(X_i)$ , where $\operatorname {rk}(X_i)$ denotes the rank of matrix $X_i$ . The sum-rank distance between $X,Y \in \mathbb {F}_q^{\mathbf {n}\times \mathbf {m}}$ is $\mathrm {srk}(X - Y)$ .

Note that the sum-rank metric is indeed an instance of the projective metric: take the set $\mathcal {F}_{\text {srk}}$ containing all possible spans of a tuple of matrices, all equal to the zero matrix except for one which is a rank-one matrix. The sum-rank metric has been studied in the context of the Eigenvalue Method in [Reference Abiad, Khramova and Ravagnani5]. Abiad et al. establish various properties of the sum-rank-metric graph, which is the graph with vertex set ${\mathbb {F}}_q^{\textbf {n} \times \textbf {m}}$ , where two vertices are adjacent if their sum-rank distance equals 1. Besides, new bounds on the maximum cardinality of sum-rank-metric codes are derived using the Ratio-type bound. These new bounds improve on the state-of-the-art bounds for several choices of the parameters.

For specific choices of set $\mathcal {F}$ , we want to be able to compare the results of the Eigenvalue Method to state-of-the-art bounds. Depending on the set $\mathcal {F}$ , bounds may exist for the specific metric arising in that case, like for the sum-rank metric. However, a bound for general codes in the projective metric also exists, namely, a Singleton-type bound.

Theorem 4.11 (Singleton-type bound, [Reference Riccardi and Sauerbier Couvée46, Theorem 83])

Let $\mathcal {C} \subseteq {\mathbb {F}}_q^n$ be a code of minimum projective $\mathcal {F}$ -distance d. For $t \in \{0,\ldots , n\}$ define $\mu _{\mathcal {F}}(t)$ as the maximum cardinality of a subset $\mathcal {G} \subseteq \mathcal {F}$ such that:

• • all $\textbf {f}_i \in \mathcal {G}$ are linearly independent over ${\mathbb {F}}_q$ ;

• • all $\textbf {v} \in \langle \mathcal {G} \rangle $ have $w_{\mathcal {F}}(\textbf {v}) \leq t$ .

Then,

$$\begin{align*}|\mathcal{C}| \leq q^{n-\mu_{\mathcal{F}}(d-1)} \leq q^{n-d+1}. \end{align*}$$

4.3 Phase-rotation metric

Another example of a projective metric is the phase-rotation metric. Note that although this is an instance of the previous metric, the phase-rotation metric is treated separately since we go into more depth with this metric.

The phase-rotation metric, which was introduced in [Reference Gabidulin and Bossert31], is particularly suitable for decoding in a binary channel where errors are caused by phase inversions (where all zeros change to ones and all ones change to zeros), and random bit errors (where at random, a zero changes to a one or a one changes to a zero).

Example 4.12 Consider the instance where $n=4, q=2$ . Take $\textbf {x}=(0,0,0,0)$ , $\textbf {y}=(1,0,0,1),$ and $\textbf {z}=(1,1,0,1)$ , which are vectors in $\mathbb {F}_2^4$ . Then, the phase-rotation distance between $\textbf {x}$ and $\textbf {y}$ is 2 since the vectors differ in two coordinates and

$$\begin{align*}\textbf{x} - \textbf{y} = (0,0,0,0)-(1,0,0,1) = (1,0,0,1) = \textbf{e}_1 + \textbf{e}_4. \end{align*}$$

The phase-rotation distance between $\textbf {x}$ and $\textbf {z}$ is also 2, even though the vectors differ in three coordinates, because

$$\begin{align*}\textbf{x} - \textbf{z} = (0,0,0,0)-(1,1,0,1)=(1,1,0,1) = (1,1,1,1)+(0,0,1,0) = \textbf{1} + \textbf{e}_3. \end{align*}$$

Now, we apply the Eigenvalue Method to the discrete metric space $({\mathbb {F}}_q^n, d_{\text {pr}})$ with the phase-rotation distance. Define the phase-rotation distance graph $G_{\text {pr}}({\mathbb {F}}_q^n)$ as the graph with vertex set ${\mathbb {F}}_q^n$ , where vertices $\textbf {x},\textbf {y} \in {\mathbb {F}}_q^n$ are adjacent if $d_{\text {pr}}(\textbf {x},\textbf {y})=1$ . Note that this is exactly the projective $\mathcal {F}$ -distance graph for the specific set $\mathcal {F}=\mathcal {F}_{\text {pr}}$ . Next, we check the conditions of the Eigenvalue Method. Since the phase-rotation distance graph equals the projective $\mathcal {F}$ -distance graph $G_{\mathcal {F}}({\mathbb {F}}_q^n)$ for $\mathcal {F} = \mathcal {F}_{\text {pr}}$ , condition (C1) and properties (P1) and (P2) follow immediately.

So condition (C1) and properties (P1) and (P2) are met. Next, we check property (P3). Distance-regularity does not follow immediately like the other properties of $G_{\text {pr}}({\mathbb {F}}_q^n)$ , but the following result gives a necessary and sufficient condition for distance-regularity of $G_{\text {pr}}({\mathbb {F}}_q^n)$ .

Proof ( $\Leftarrow $ ) We prove the three cases, $n=1$ , $n=2,$ and $q=2$ , separately. The case $n=1$ follows immediately: the graph for $n=1$ is a complete graph on q vertices, which is distance-regular.

When $q=2$ , the phase-rotation distance graph equals the folded cube graph of dimension $n+1$ , which is a distance-regular graph [Reference Brouwer, Cohen and Neumaier15, Section 9.2D].

Consider the case $n=2$ , $q \geq 3$ . The diameter of $G_{\text {pr}}({\mathbb {F}}_q^2)$ then equals 2. Observe that it suffices to show that for vertices $\textbf {u}$ and $\textbf {v}$ with $d(\textbf {u},\textbf {v})=i$ the values $b_i(\textbf {u},\textbf {v})$ and $c_i(\textbf {u},\textbf {v})$ , the number of neighbors of $\textbf {u}$ at distance $i+1$ , $i-1$ from $\textbf {v,}$ respectively, do not depend on the choice of $\textbf {u}$ and $\textbf {v}$ for $i=0,1,2$ . Since the phase-rotation distance graph is vertex-transitive, we may assume without loss of generality that $\textbf {v}=\textbf {0}$ . Since $G_{\text {pr}}({\mathbb {F}}_q^2)$ is $3(q-1)$ -regular, we have $b_0 = 3(q-1)$ which does not depend on the choice of $\textbf {u}$ . Also $c_0=0$ , $c_1=1$ , and $b_2=0$ by definition. Now, consider $b_1(\textbf {u},\textbf {v})$ . A vertex $\textbf {u}$ at distance $1$ from $\textbf {v}=\textbf {0}$ has the form $(a,0)$ , $(0,a),$ or $(a,a)$ with $a \in {\mathbb {F}}_q^*$ . The neighbors of $\textbf {v}$ at distance $2$ from $(a,0)$ , $(0,a)$ , $(a,a)$ with $a \in {\mathbb {F}}_q^*$ are

$$\begin{align*}\left\{(0,x), (y,y): x,y \in {\mathbb{F}}_q^*, x \neq -a,y \neq a \right\}, \ \left\{(x,0), (y,y): x,y \in {\mathbb{F}}_q^*, x \neq -a,y \neq a \right\}, \end{align*}$$

$$\begin{align*}\left\{(0,x), (y,0): x,y \in {\mathbb{F}}_q^*, x,y \neq a \right\}, \end{align*}$$

respectively. All these three sets contain $2q-4$ distinct vertices so $b_1 = 2q-4$ , which is independent of the choice of $\textbf {u}$ . Now, consider $c_2(\textbf {u},\textbf {v})$ . A vertex $\textbf {u}$ at distance $2$ from $\textbf {v}=\textbf {0}$ has the form $(a,b)$ with $a,b \in {\mathbb {F}}_q^*, a \neq b$ . The neighbors of $\textbf {v}$ that are also neighbors $(a,b)$ with $a,b \in {\mathbb {F}}_q^*, a \neq b$ are the vertices

$$\begin{align*}\{(0,b), (a,0), (a,a), (b,b), (0, b-a), (a-b,0)\}. \end{align*}$$

These are six distinct vertices, independent of the choice of $a,b \in {\mathbb {F}}_q^*$ such that $a \neq b$ , so independent of the choice of $\textbf {u}$ , and hence $c_2=6$ . This proves that $G_{\text {pr}}({\mathbb {F}}_q^2)$ for $q \geq 3$ is distance-regular.

( $\Rightarrow $ ) Now, we show that $n=1$ , $n=2,$ and $q=2$ are the only cases where the phase-rotation distance graph is distance-regular. Let $n \geq 3$ and $q \geq 3$ . Define $r := \lceil \frac {n}{2} \rceil $ . Let

$$\begin{align*}\textbf{x} := (\underbrace{1,\ldots, 1}_{r-1 \text{ times}}, a, 0, \ldots,0), \qquad \textbf{y} := (\underbrace{1, \ldots, 1}_{r \text{ times}},0, \ldots, 0), \end{align*}$$

for some fixed $a \in {\mathbb {F}}_q^*, a \neq 1$ . Note that $d_{\text {pr}}(\textbf {x},\textbf {0}) = d_{\text {pr}}(\textbf {y},\textbf {0}) = r$ .

When n is odd, consider $c_r$ . For $\textbf {x}$ and the zero vector, we find

$$\begin{align*}c_r(\textbf{x},\textbf{0}) = p_{1, r-1}^r(\textbf{x},\textbf{0}) = r, \end{align*}$$

since the only neighbors of $\textbf {x}$ at distance $r-1$ from the zero vector are the vectors where one of the nonzero coordinates of $\textbf {x}$ is replaced with a zero. For $\textbf {y}$ and the zero vector, we find

$$\begin{align*}c_r(\textbf{y},\textbf{0}) = p_{1, r-1}^r(\textbf{y},\textbf{0}) \geq r+1, \end{align*}$$

since $\textbf {y}$ has r neighbors at distance $r-1$ from the zero vector similarly as $\textbf {x}$ and the zero vector, but $\textbf {y}$ also has the vector starting with $r+1$ ones and then $r-2$ zeros, which is at distance $r-1$ from the zero vector, as a neighbor. So the number $c_r$ for n odd depends on the choice of vertices.

When n is even, we consider $a_r$ . For $\textbf {x}$ and the zero vector, we get:

$$\begin{align*}a_r(\textbf{x},\textbf{0}) = p_{1,r}^r(\textbf{x},\textbf{0}) = (q-2)r, \end{align*}$$

since the only neighbors of $\textbf {x}$ at distance r from the zero vector are the vectors where one of the r nonzero coordinates of $\textbf {x}$ is replaced by another nonzero element of ${\mathbb {F}}_q$ . For $\textbf {y}$ and the zero vector, we have

$$\begin{align*}a_r(\textbf{y},\textbf{0})=p_{1,r}^r(\textbf{y},\textbf{0}) \geq (q-2)r+1, \end{align*}$$

since $\textbf {y}$ has $(q-2)r$ neighbors at distance r from the zero vector similarly as $\textbf {x}$ and the zero vector, but $\textbf {y}$ also has the vector starting with $r+1$ ones and then $r-1$ zeros, which is at distance r from the zero vector, as neighbor. So the value of $a_r$ depends on the choice of vertices for n even. Hence, $G_{\text {pr}}({\mathbb {F}}_q^n)$ is not distance-regular when $n \geq 3, q \geq 3$ , which proves the result.

The latter result shows that property (P3) is met when $n \geq 3$ and $q \geq 3$ . Moreover, since the graph $G_{\text {pr}}({\mathbb {F}}_q^n)$ has the desired properties (P1) and (P2), both the Inertia-type bound and the Ratio-type bound can be applied to this graph. To do so, we first need to determine the adjacency eigenvalues of the phase-rotation distance graph.

Proposition 4.16 The adjacency eigenvalues of $G_{\text {pr}}({\mathbb {F}}_q^n)$ for $n \geq 2$ are

for every tuple $\textbf {r}= (r_1, \ldots , r_n) \in [\![q-1]\!]^n$ .

Note that denotes the indicator function.

For the proof of Proposition 4.16, we use characters of groups. So we first present some background on characters and how they can be used to determine the eigenvalues of a Cayley graph. For more details about the latter, we refer the reader to [Reference Lovász41].

Example 4.18 gives the characters of cyclic groups. Note that any finite abelian group G is isomorphic to a product of cyclic groups, i.e., $G \cong \mathbb {Z}/m_1 \mathbb {Z} \times \cdots \times \mathbb {Z}/m_l \mathbb {Z}$ . It turns out that the characters of a Cartesian product of groups are related to the characters of the individual groups.

Proof Let $*, *'$ denote the operation in G, respectively, H. Let $(g_1,h_1), (g_2,h_2) \in G \times H$ . We want to prove that $\chi _{i,j}((g_1,h_1)(* \times *')(g_2,h_2)) = \chi _{i,j}((g_1,g_2)) \chi _{i,j}((g_2,h_2))$ since this is a sufficient condition for $\chi _{i,j}$ to be a group homomorphism. Observe:

$$\begin{align*}\chi_{i,j}((g_1,h_1)(* \times *')(g_2,h_2)) &= \chi_{i,j}((g_1*g_2,h_1*'h_2)) \\ &= \chi_{G,i}(g_1*g_2) \chi_{H,j}(h_1*'h_2) \end{align*}$$

and

$$\begin{align*}\chi_{i,j}((g_1,g_2)) \chi_{i,j}((g_2,h_2)) &= \chi_{G,i}(g_1)\chi_{H,j}(g_2)\chi_{G,i}(h_1)\chi_{H,j}(h_2) \\ &= \chi_{G,i}(g_1*g_2) \chi_{H,j}(h_1*'h_2), \end{align*}$$

since $\chi _{G,i}$ and $\chi _{H,j}$ are group homomorphisms of $G, H$ , respectively. So $\chi _{i,j}$ is a group homomorphism of $G \times H$ . Moreover, for any $(g,h) \in G \times H$ , we have

$$\begin{align*}|\chi_{i,j}((g,h))| = |\chi_{G,i}(g)| \cdot |\chi_{H,j}(h)| = 1, \end{align*}$$

since $\chi _{G,i}$ and $\chi _{H,j}$ are characters of $G,H,$ respectively. Hence, $\chi _{i,j}$ is a character of $G \times H$ .

With Lemma 4.19 and Example 4.18, the characters of finite abelian groups are now completely defined. The next result tells us how characters determine the adjacency eigenvalues of a Cayley graph, which is the final bit of information needed for the proof of Proposition 4.16.

Lemma 4.20 [Reference Lovász41]

Let G be a finite abelian group, let $\chi _i$ be the characters of G, and let $S \subseteq G$ be a symmetric set. The adjacency eigenvalues of the Cayley graph over group G with connecting set S are given by

$$\begin{align*}\lambda_i = \sum_{s \in S} \chi_i(s), \end{align*}$$

for $i = 0, \ldots , |G|-1$ .

Proof of Proposition 4.16.

The graph $G_{\text {pr}}({\mathbb {F}}_q^n)$ is a Cayley graph over ${\mathbb {F}}_q^n$ with connecting set $S := \{c\textbf {x}: c \in {\mathbb {F}}_q^*, \textbf {x} \in \{\textbf {e}_1, \ldots , \textbf {e}_n, \textbf {1}\} \}$ . First, we determine the characters of ${\mathbb {F}}_q$ . The field ${\mathbb {F}}_q$ seen as a group is isomorphic to $(\mathbb {Z}/p\mathbb {Z})^k$ for some prime p such that $q=p^k$ . Since the characters of $\mathbb {Z}/p\mathbb {Z}$ are known to be $\chi _r(x) = (\zeta _p)^{rx}$ for $r=0, \ldots , p-1,$ where $\zeta _p = \exp \left ( {2 \pi i}/{p} \right )$ (see Example 4.18), Lemma 4.19 tells us that the characters of ${\mathbb {F}}_q$ are

$$\begin{align*}\chi_{\textbf{r}}(\textbf{x}) = \prod_{j=1}^k (\zeta_p)^{r_jx_j} = (\zeta_p)^{\sum_{j=1}^k r_jx_j}, \end{align*}$$

for $\textbf {r} \in [\![p-1]\!]^k$ , where $\textbf {x} = (x_1, \ldots , x_k) \in (\mathbb {Z}/p\mathbb {Z})^k \cong {\mathbb {F}}_q$ . The multiplication $r_jx_j$ is taken modulo p; this abuse of notation is used more often in this proof. Now, we can determine the characters of ${\mathbb {F}}_q^n$ , again using Lemma 4.19:

(4.1) $$ \begin{align} \chi_{\textbf{r}}((\textbf{x}_1, \ldots, \textbf{x}_n)) = \prod_{l=1}^n \chi_{\textbf{r}_l}(\textbf{x}_l) = \prod_{l=1}^n (\zeta_p)^{\sum_{j=1}^k r_{l_j}x_{l_j}} = (\zeta_p)^{\sum_{l=1}^n \sum_{j=1}^k r_{l_j}x_{l_j}}, \end{align} $$

for $\textbf {r} = (\textbf {r}_1, \ldots , \textbf {r}_n)$ with $\textbf {r}_l = (r_{l_1}, \ldots , r_{l_k}) \in [\![p-1]\!]^k$ , $l=1, \ldots , n$ , where $\textbf {x}_l \in (\mathbb {Z}/p\mathbb {Z})^k \cong {\mathbb {F}}_q$ , $l=1, \ldots , n$ . By Lemma 4.20, the adjacency eigenvalues of $G_{\text {pr}}({\mathbb {F}}_q^n)$ are then

$$\begin{align*}\lambda_{\textbf{r}} = \sum_{\textbf{s} \in S} \chi_{\textbf{r}}(\textbf{s}), \end{align*}$$

for tuples $\textbf {r} \in ([\![p-1]\!]^k)^n$ , where every $\textbf {s} \in S$ is viewed as an element of $([\![p-1]\!]^k)^n$ .

Now, we simplify this expression. Let $\textbf {s} \in S$ , then $\textbf {s} = c \textbf {x}$ for some $c \in {\mathbb {F}}_q^*$ and $\textbf {x} \in \{\textbf {e}_1, \ldots , \textbf {e}_n, \textbf {1}\}$ . If $\textbf {s}$ is viewed as an element of $([\![p-1]\!]^k)^n$ , then $\textbf {s} = (\textbf {c}, \textbf {0}, \ldots , \textbf {0}), \ldots , (\textbf {0}, \ldots , \textbf {0}, \textbf {c})$ or $(\textbf {c}, \ldots , \textbf {c})$ for some $\textbf {c} \in [\![p-1]\!]^k$ , $\textbf {c} \neq \textbf {0}$ . So for a fixed $\textbf {r} \in ([\![p-1]\!]^k)^n$ , we get:

$$\begin{align*}\lambda_{\textbf{r}} = \sum_{\textbf{c} \in [\![p-1]\!]^k, \textbf{c} \neq \textbf{0}} \chi_{\textbf{r}}((\textbf{c}, \textbf{0}, \ldots, \textbf{0})) + \cdots + \chi_{\textbf{r}}((\textbf{0}, \ldots, \textbf{0}, \textbf{c})) + \chi_{\textbf{r}}((\textbf{c}, \ldots, \textbf{c})). \end{align*}$$

Since $\chi _{r_l}(\textbf {0)} = 1$ for $l=1, \ldots , n$ , this simplifies to

$$\begin{align*}\lambda_{\textbf{r}} = \sum_{\textbf{c} \in [\![p-1]\!]^k, \textbf{c} \neq \textbf{0}} \chi_{\textbf{r}_1}(\textbf{c}) + \cdots + \chi_{\textbf{r}_n}(\textbf{c}) + \prod_{l=1}^n \chi_{\textbf{r}_l}(\textbf{c}). \end{align*}$$

Using the expression for the characters from Equation (4.1) and letting the sum also run over $\textbf {c}=\textbf {0}$ gives:

(4.2) $$ \begin{align} \lambda_{\textbf{r}} = \sum_{\textbf{c} \in [\![p-1]\!]^k} \sum_{l=1}^n (\zeta_p)^{\sum_{j=1}^k r_{l_j}c_j} + (\zeta_p)^{\sum_{l=1}^n \sum_{j=1}^k r_{l_j} c_j} - n - 1. \end{align} $$

We know that $1+ \zeta _m + \cdots + (\zeta _m)^{m-1} = 0$ for any m-th root of unity $\zeta _m \neq 1$ . Since $\sum _{j=1}^k r_{l_j}c_j\ \mod p$ attains every value of $\{0, \ldots , p-1\}$ equally often when $\textbf {r}_l \neq \textbf {0}$ for $\textbf {c} \in [\![p-1]\!]^k$ , we get

$$\begin{align*}\sum_{\textbf{c} \in [\![p-1]\!]^k} (\zeta_p)^{\sum_{j=1}^k r_{l_j}c_j} = 0 \end{align*}$$

for $l=1, \ldots , n$ if $\textbf {r}_l \neq \textbf {0}$ . If $\textbf {r}_l = \textbf {0}$ , then

$$\begin{align*}\sum_{\textbf{c} \in [\![p-1]\!]^k} (\zeta_p)^{\sum_{j=1}^k r_{l_j}c_j} = \sum_{\textbf{c} \in [\![p-1]\!]^k} 1 = p^k = q. \end{align*}$$

Also $\sum _{l=1}^n \sum _{j=1}^k r_{l_j} c_j\ \mod p$ attains every value in $\{0,\ldots , p-1\}$ equally often when $\sum _{l=1}^n \textbf {r}_l = (\sum _{l=1}^n r_{l_1}, \ldots , \sum _{l=1}^n r_{l_k}) \not \equiv \textbf {0}\ \mod p$ for $\textbf {c} \in [\![p-1]\!]^k$ . So

$$\begin{align*}\sum_{\textbf{c} \in [\![p-1]\!]^k} (\zeta_p)^{\sum_{l=1}^n \sum_{j=1}^k r_{l_j} c_j} = 0 \end{align*}$$

if $\sum _{l=1}^n \textbf {r}_l \not \equiv \textbf {0}\ \mod p$ . If $\sum _{l=1}^n \textbf {r}_l \equiv \textbf {0}\ \mod p$ , then

$$\begin{align*}\sum_{\textbf{c} \in [\![p-1]\!]^k} (\zeta_p)^{\sum_{l=1}^n \sum_{j=1}^k r_{l_j} c_j} = \sum_{\textbf{c} \in [\![p-1]\!]^k} (\zeta_p)^{\sum_{j=1}^k 0 \cdot c_j} = \sum_{\textbf{c} \in [\![p-1]\!]^k} 1 = p^k = q. \end{align*}$$

Combining these four observations with Equation (4.2) gives the following formula for the eigenvalues:

for $\textbf {r}=(\textbf {r}_1, \ldots , \textbf {r}_n) \in ([\![p-1]\!]^k)^n$ .

For the last step of this proof, we note that $(a_1, \ldots , a_k) \in [\![p-1]\!]^k$ can be related to $a \in [\![q-1]\!]$ , where $q=p^k$ , by $a = \sum _{j=1}^k a_j p^{j-1}$ . Then, the conditions $\textbf {r}_l = \textbf {0}$ and $\sum _{l=1}^n \textbf {r}_l \equiv \textbf {0}\ \mod p$ are equivalent to the conditions $r_l = 0$ and $\sum _{l=1}^n r_l \equiv 0\ \mod q$ , respectively. Using this observation, the eigenvalues of $G_{\text {pr}}({\mathbb {F}}_q^n)$ for $n \geq 2$ equal

for tuples $\textbf {r}=(r_1, \ldots , r_n) \in [\![q-1]\!]^n$ , which is what we wanted to prove.

Now, we can derive the distinct eigenvalues of the phase-rotation distance graph for $n \geq 2$ ; these distinct eigenvalues follow directly from Proposition 4.16.

Corollary 4.21 The distinct adjacency eigenvalues of $G_{\text {pr}}({\mathbb {F}}_q^n)$ for $n \geq 2$ are

$$\begin{align*}\begin{cases} 2i - n -1, \quad \text{for } i=1,3, \ldots, n-1, n+1 &\text{if } q=2, n \text{ even}, \\ 2i-n-1, \quad \text{for } i=0, 2, \ldots, n-1, n+1 &\text{if } q=2, n \text{ odd}, \\ iq - n - 1, \quad \text{for } i=0, 1, \ldots, n-1, n+1 &\text{if } q \geq 3. \\ \end{cases} \end{align*}$$

The expressions for the (distinct) eigenvalues of the phase-rotation adjacency graph can be used in the Inertia-type bound and the Ratio-type bound to derive new bounds on the cardinality of phase-rotation codes. Then, we compare these new bounds to a state-of-the-art bound, namely, a Singleton-type bound for phase-rotation codes.

Theorem 4.22 (Singleton-type bound, [Reference Riccardi and Sauerbier Couvée46])

Let $\mathcal {C} \subseteq {\mathbb {F}}_q^n$ be a code of minimum phase-rotation distance d. Then,

$$\begin{align*}|\mathcal{C}| \leq \begin{cases} q^{n-d+1} &\text{if } d<1+\lceil n-\tfrac{n}{q} \rceil, \\ 1 &\text{otherwise}. \end{cases} \end{align*}$$

This bound follows from the Singleton-type bound for the projective metric from Theorem 4.11 by using the result of [Reference Riccardi and Sauerbier Couvée46, Example 81], which states that

$$\begin{align*}w_{\mathcal{F}_{\text{pr}}}(t) = \begin{cases} t &\text{if } t< \lceil n -\tfrac{n}{q} \rceil, \\ n &\text{otherwise}, \end{cases} \end{align*}$$

for the set $\mathcal {F}_{\text {pr}}$ as defined in Definition 4.4.

We start by considering the Ratio-type bound on the k-independence number for $k=1,2,3$ , since there are explicit expressions for this bound that are independent of a choice of polynomial $p \in {\mathbb {R}}_k[x]$ (see [Reference Haemers34, Theorem 3.2], Theorem 3.4, and Theorem 3.5, respectively). First, consider the ratio bound on the independence number $\alpha $ , i.e., the Ratio-type bound on the k-independence number $\alpha _k$ for $k=1$ . Applied to the phase-rotation distance graph $G_{\text {pr}}({\mathbb {F}}_q^n)$ , this ratio bound gives the following upper bound on the independence number $\alpha (G_{\text {pr}}({\mathbb {F}}_q^n))$ .

Theorem 4.23 Let $n \geq 2$ . Then,

$$\begin{align*}\alpha(G_{\text{pr}}({\mathbb{F}}_q^n)) \leq \begin{cases} 2^{n-1} \frac{n-1}{n} & \text{if } q=2, n \text{ even}, \\ q^{n-1} &\text{if } q=2, n \text{ odd or } q \geq 3. \end{cases} \end{align*}$$

Proof The largest eigenvalue is $\lambda _1 = (n+1)(q-1)$ for all $q,n$ , while the smallest eigenvalue is $\lambda _{q^n} = 1-n$ if $q=2$ , n even and $\lambda _{q^n} = -n-1$ otherwise. The ratio bound from [Reference Haemers34, Theorem 3.2] is applicable, so for $q=2$ , n even, we get

$$\begin{align*}\alpha(G_{\text{pr}}(\mathbb{F}_2^n)) \leq 2^n \frac{-(1-n)}{(n+1)-(1-n)} = 2^{n-1} \frac{n-1}{n}. \end{align*}$$

For $q=2$ , n odd or $q \geq 3$ , we obtain

$$\begin{align*}\alpha(G_{\text{pr}}({\mathbb{F}}_q^n)) \leq q^n \frac{-(-n-1)}{(n+1)(q-1)-(-n-1)} = q^n \frac{n+1}{q(n+1)} = q^{n-1}.\\[-41pt] \end{align*}$$

Let $A_q^{\text {pr}}(n,d)$ denote the maximum cardinality of code in ${\mathbb {F}}_q^n$ with minimum phase-rotation distance d. The upper bounds from Theorem 4.23 can be translated to upper bounds on $A_q^{\text {pr}}(n,d)$ via Lemma 3.1.

Corollary 4.24 The cardinality of phase-rotation codes in ${\mathbb {F}}_q^n$ of minimum distance 2 with $n \geq 2$ is upper bounded by:

(4.3) $$ \begin{align} A_q^{\text{pr}}(n,2) \leq \begin{cases} 2^{n-1} \frac{n-1}{n} & \text{if } q=2, n \text{ even}, \\ q^{n-1} &\text{if } q=2, n \text{ odd or } q \geq 3. \end{cases} \end{align} $$

Next, we compare these upper bounds from Corollary 4.24 to the Singleton-type upper bound from Theorem 4.22.

So the Ratio-type bound on the k-independence number for $k=1$ gives a bound on the maximum cardinality of phase-rotation codes that is at least as good as the Singleton-type bound. Next, we consider the Ratio-type bound on the $2$ -independence number $\alpha _2$ . The cases $q=2$ and $q \geq 3$ are treated separately since the expression for the distinct eigenvalues of $G_{\text {pr}}({\mathbb {F}}_q^n)$ for these cases is sufficiently different. First, consider the case $q=2$ (and $k=2$ ).

Theorem 4.26 Let $n \geq 3$ . Then,

$$\begin{align*}\alpha_2(G_{\text{pr}}(\mathbb{F}_2^n)) \leq \begin{cases} 2^n \frac{n-2}{n(n+4)} & \text{if } n \equiv 0\quad \mod 4, \\2^n \frac{n-3}{(n+3)(n-1)} & \text{if } n \equiv 1\quad \mod 4, \\2^n \frac{1}{n+2} & \text{if } n \equiv 2\quad \mod 4, \\2^n \frac{1}{n+5} & \text{if } n \equiv 3\quad \mod 4. \end{cases} \end{align*}$$

Proof Since $n \geq 3$ , $G_{\text {pr}}(\mathbb {F}_2^n)$ has at least three distinct eigenvalues, so Theorem 3.4 is applicable. The largest eigenvalue which is at most $-1$ satisfies:

$$\begin{align*}2i-n-1 \leq -1 \Leftrightarrow 2i \leq n \Leftrightarrow i \leq \frac{n}{2}. \end{align*}$$

We start with the case where n is even, or $n \equiv 0,2\ \mod 4$ . Since i has to be odd for $2i-n-1$ to be an eigenvalue when n is even, we get $i = \frac {n}{2}$ if $n \equiv 2\ \mod 4$ and $i = \frac {n}{2}-1$ if $n \equiv 0\ \mod 4$ . If $n \equiv 2\ \mod 4$ , we have $\theta _i = -1, \theta _{i-1} = 3, \theta _0 = n+1$ . Then,

$$\begin{align*}\alpha_2(G_{\text{pr}}(\mathbb{F}_2^n)) \leq 2^n \frac{n+1-3}{(n+2)(n-2)} = \frac{2^n}{n+2}. \end{align*}$$

If $n \equiv 0\ \mod 4$ , we have $\theta _i = -3, \theta _{i-1} = 1, \theta _0 = n+1$ . Then,

$$\begin{align*}\alpha_2(G_{\text{pr}}(\mathbb{F}_2^n)) \leq 2^n \frac{n+1-3}{(n+4)n} = 2^n \frac{n-2}{n(n+4)}. \end{align*}$$

Next, we deal with the case where n is odd, or $n \equiv 1,3\ \mod 4$ . Since i has to be even for $2i-n-1$ to be an eigenvalue when n is odd, we get $i = \frac {n-1}{2}$ if $n \equiv 1\ \mod 4$ and $i = \frac {n-1}{2}-1$ if $n \equiv 3\ \mod 4$ . If $n \equiv 1\ \mod 4$ , we have $\theta _i = -2, \theta _{i-1} = 2, \theta _0 = n+1$ . Then,

$$\begin{align*}\alpha_2(G_{\text{pr}}(\mathbb{F}_2^n)) \leq 2^n \frac{n+1-4}{(n+3)(n-1)} = 2^n \frac{n-3}{(n+3)(n-1)}. \end{align*}$$

If $n \equiv 3\ \mod 4$ , we have $\theta _i = -4, \theta _{i-1} = 0, \theta _0 = n+1$ . Then,

$$ \begin{align*} &\alpha_2(G_{\text{pr}}(\mathbb{F}_2^n)) \leq 2^n \frac{n+1}{(n+5)(n+1)} = \frac{2^n}{n+5}. \end{align*} $$

The upper bounds from Theorem 4.26 can be translated to upper bounds on $A^{\text {pr}}_2(n,3)$ via Lemma 3.1.

Corollary 4.27 The maximum cardinality of phase-rotation codes in $\mathbb {F}_2^n$ of minimum distance 3 with $n \geq 3$ is upper bounded by

(4.4) $$ \begin{align} A_2^{\text{pr}}(n,3) \leq \begin{cases} 2^n \frac{n-2}{n(n+4)} & \text{if } n \equiv 0\quad \mod 4, \\ 2^n \frac{n-3}{(n+3)(n-1)} & \text{if } n \equiv 1\quad \mod 4, \\ 2^n \frac{1}{n+2} & \text{if } n \equiv 2\quad \mod 4, \\ 2^n \frac{1}{n+5} & \text{if } n \equiv 3\quad \mod 4. \end{cases} \end{align} $$

Now, we compare these upper bounds from Corollary 4.27 to the Singleton-type upper bound from Theorem 4.22.

Proof The upper bound of the Singleton-type bound for $d=3$ and $q=2$ is $2^{n-2}$ if $3<1+ \lceil n-\tfrac {n}{2} \rceil $ , which is exactly if $\tfrac {n}{2}> 2 \Leftrightarrow n>4$ . If this is the case, then we can compare the bounds and determine when the bounds from Equation (4.4) are smaller than or equal to $2^{n-2}$ . For $n \equiv 0\ \mod 4,$ we have

$$\begin{align*}2^n \frac{n-2}{n(n+4)} \leq 2^{n-2} \Leftrightarrow 2^2(n-2) \leq n(n+4) \Leftrightarrow n^2 \geq -8, \end{align*}$$

which trivially holds true. For $n \equiv 1\ \mod 4,$

$$\begin{align*}2^n \frac{n-3}{(n+3)(n-1)} &\leq 2^{n-2} \Leftrightarrow 2^2(n-3) \\ &\leq (n+3)(n-1) \Leftrightarrow n^2-2n+9 = (n-1)^2+8 \geq 0. \end{align*}$$

Also this holds true. For $n \equiv 2\ \mod 4$ ,

$$\begin{align*}2^n \frac{1}{n+2} \leq 2^{n-2} \Leftrightarrow 2^2 \leq n+2 \Leftrightarrow n \geq 2, \end{align*}$$

which is true by the assumption on n. Lastly, for $n \equiv 3\ \mod 4,$ we get:

$$\begin{align*}2^n \frac{1}{n+5} \leq 2^{n-2} \Leftrightarrow 2^2 \leq n+5 \Leftrightarrow n \geq -1, \end{align*}$$

which also holds by the assumption on n.

Next, we consider the case that $n=3,4$ . Then, the Singleton-type bound gives an upper bound of $1$ , while our bounds give values of $2^3 \cdot \tfrac {1}{8} = 1$ and $2^4 \cdot \tfrac {2}{32}=1$ for $n=3,4,$ respectively. Hence, the upper bounds on $A_2^{\text {pr}}(n,3)$ from Equation (4.4) are no worse than the Singleton-type upper bound from Theorem 4.22.

Hence, the Ratio-type bound gives upper bounds on the cardinality of phase-rotation codes that are at least as good as the Singleton-type bound for $k=2$ and $q=2$ . Next, we consider the case $k=2$ and $q \geq 3$ .

Theorem 4.29 Let $n \geq 2$ and $q \geq 3$ . Then,

$$\begin{align*}\alpha_2(G_{\text{pr}}({\mathbb{F}}_q^n)) \leq q^{n-2} \frac{n(n+1)+\lfloor \tfrac{n}{q} \rfloor q \big( -2-2n+q+ \lfloor \tfrac{n}{q} \rfloor q \big) }{ \big( n- \lfloor \tfrac{n}{q} \rfloor \big) \big( n+1-\lfloor \tfrac{n}{q} \rfloor \big)}. \end{align*}$$

Proof The distinct eigenvalues of $G_{\text {pr}}({\mathbb {F}}_q^n)$ for $n\geq 2$ and $q \geq 3$ are $iq-n-1$ for $i=0,1\ldots , n-1,n+1$ . Since $n \geq 2$ , we have at least three distinct eigenvalues, so Theorem 3.4 is applicable. First, we determine the largest eigenvalue which is at most $-1$ :

$$\begin{align*}iq-n-1 \leq -1 \Leftrightarrow iq \leq n \Leftrightarrow i \leq \frac{n}{q}. \end{align*}$$

Taking $i = \lfloor \frac {n}{q} \rfloor $ gives this eigenvalue. Note that $0 \leq \lfloor \frac {n}{q} \rfloor \leq \frac {n}{3} \leq n-1$ , so $i = \lfloor \frac {n}{q} \rfloor $ indeed gives an eigenvalue. Using the notation of Theorem 3.4, we have

$$\begin{align*}\theta_0 = (n+1)(q-1), \quad \theta_{i-1} = \left( \lfloor \tfrac{n}{q} \rfloor +1 \right)q-n-1, \quad \theta_i = \lfloor \tfrac{n}{q} \rfloor q -n-1. \end{align*}$$

Then, we obtain the following upper bound for $\alpha _2(G_{\text {pr}}({\mathbb {F}}_q^n))$ :

$$\begin{align*}q^n \frac{(n+1)(q-1) + \left( \lfloor \tfrac{n}{q} \rfloor q -n-1 \right) \left( ( \lfloor \tfrac{n}{q} \rfloor +1 )q-n-1 \right)}{\left( (n+1)(q-1) - (\lfloor \tfrac{n}{q} \rfloor q -n-1) \right) \left( (n+1)(q-1) - \big( ( \lfloor \tfrac{n}{q} \rfloor +1 ) q-n-1 \big) \right)} \end{align*}$$

$$\begin{align*}= q^{n-2} \frac{n(n+1)+ \lfloor \tfrac{n}{q} \rfloor q \big( -2-2n+q+ \lfloor \tfrac{n}{q} \rfloor q \big)}{\big( n-\lfloor \tfrac{n}{q} \rfloor \big) \big( n+1-\lfloor \tfrac{n}{q} \rfloor \big)}.\\[-41pt] \end{align*}$$

This upper bound from Theorem 4.29 can be translated to an upper bound on $A_q^{\text {pr}}(n,3)$ via Lemma 3.1.

Corollary 4.30 The cardinality of phase-rotation codes of minimum distance $3$ with $q \geq 3$ and $n \geq 2$ is upper bounded by

(4.5) $$ \begin{align} A_q^{\text{pr}}(n,3) \leq q^{n-2} \frac{n(n+1)+ \lfloor \tfrac{n}{q} \rfloor q \big( -2-2n+q+ \lfloor \tfrac{n}{q} \rfloor q \big)}{\big( n-\lfloor \tfrac{n}{q} \rfloor \big) \big( n+1-\lfloor \tfrac{n}{q} \rfloor \big)}. \end{align} $$

A comparison of this upper bound from Corollary 4.30 with the Singleton-type bound from Theorem 4.22 gives the following result.

Proof The Singleton-type bound for $d=3$ is $q^{n-2}$ if $3<1+\lceil n - \tfrac {n}{q} \rceil $ , which happens exactly if $n - \tfrac {n}{q}>2 \Leftrightarrow n > 2+\tfrac {2}{q-1}$ . If $q=3$ , then we need $n>3$ , and if $q \geq 4$ , then $n \geq 3$ suffices. In these cases, we prove that the upper bound from Equation (4.5) is at most $q^{n-2}$ .

If $n<q$ , then $\lfloor \tfrac {n}{q} \rfloor = 0$ and the upper bound from the Ratio-type bound reduces to

$$\begin{align*}A_q^{\text{pr}}(n,3) \leq q^{n-2} \frac{n(n+1)}{n(n+1)} = q^{n-2}. \end{align*}$$

This exactly equals the upper bound from the Singleton-type bound for $d=3$ .

If $q \leq n < 2q$ , then $\lfloor \tfrac {n}{q} \rfloor = 1$ and the upper bound from the Ratio-type bound reduces to

$$\begin{align*}A_q^{\text{pr}}(n,3) \leq q^{n-2} \frac{n(n+1)+q(-2-2n+2q)}{(n-1)n}. \end{align*}$$

It can be verified with mathematical software that this is less than or equal to $q^{n-2}$ when $n \geq 2$ and $q \leq n <2q$ . So the desired result holds in this case.

If $2q \leq n <3q$ , then $\lfloor \tfrac {n}{q} \rfloor = 2$ and the upper bound from the Ratio-type bound reduces to

$$\begin{align*}A_q^{\text{pr}}(n,3) \leq q^{n-2} \frac{n(n+1)+2q(-2-2n+3q)}{(n-2)(n-1)}. \end{align*}$$

Mathematical software can show that this is less than or equal to $q^{n-2}$ if $n \geq 3$ and $2q \leq n <3q$ . Since $n \geq 2q$ and $q \geq 3$ , the condition $2q \leq n < 3q$ is actually sufficient. So also in this case, the desired result is reached.

Lastly, we consider the last case $n \geq 3q$ . We have $q \lfloor \tfrac {n}{q} \rfloor \leq n$ , so $-2-2n+q+ q \lfloor \tfrac {n}{q} \rfloor \leq -2-2n+q+n = -2-n+q < 0$ since $n \geq 3q$ . Also $q \lfloor \tfrac {n}{q} \rfloor \geq q \left ( \tfrac {n-(q-1)}{q} \right ) = n-q+1$ since $n,q$ are integral. Then,

$$\begin{align*}\lfloor \tfrac{n}{q} \rfloor q(-2-2n+q+ \lfloor \tfrac{n}{q} \rfloor q) \leq (n-q+1)(-2-n+q). \end{align*}$$

The bound from the Ratio-type bound is thus upper bounded by

$$\begin{align*}A_q^{\text{pr}}(n,3) \leq q^n \frac{n(n+1) +(n-q+1)(-2-n+q)}{(qn-n)(qn+q-n)} = q^n \frac{2+2n-q}{n(qn+q-n)}. \end{align*}$$

This is less than or equal to $q^{n-2}$ if and only if

$$\begin{align*}q^2(2+2n-q) \leq n(qn+q-n), \end{align*}$$

which can be seen to hold, using mathematical software, for $(n,q)=(9,3)$ or $n \geq 10$ and $3 \leq q \leq \frac {n}{3}$ . Since $q\geq 3$ , we have $n \geq 3q \geq 9$ . If $n=9$ , then $3 \leq q \leq \frac {n}{3} = 3$ , so $q=3$ is the only option. If $n \geq 10$ , then the desired inequality holds for $n \geq 3q$ . All in all, we also get the desired result when $n \geq 3q$ .

Next, we consider the cases where the Singleton-type bound equals $1$ . This happens if $q=3$ , $n=2,3$ or $q \geq 4$ , $n=2$ . If $n=2$ , then the bound from Equation (4.5) reduces to 1. If $q=3$ and $n=3$ , then our bound reduces to 3. So $q=3$ , $n=3$ is the only case in which the upper bound on $A_q^{\text {pr}}(n,3)$ from Equation (4.5) is worse than the upper bound from the Singleton-type bound of Theorem 4.22.

Also for $k=2$ and $q \geq 3$ bounds for phase-rotation codes that are almost always at least as good as the Singleton-type bound can be obtained from the Ratio-type bound. Now, consider the Ratio-type bound on the 3-independence number $\alpha _3$ . Again the cases $q=2$ and $q \geq 3$ are treated separately. First, the case $q=2$ (and $k=3$ ) is studied.

Theorem 4.32 Let $n \geq 5$ . Then,

$$\begin{align*}\alpha_3(G_{\text{pr}}(\mathbb{F}_2^n)) \leq \begin{cases} 2^{n-1} \frac{n^2-n+4}{n^2(n+4)} & \text{if } n \equiv 0\quad \mod 4, \\ 2^{n-1} \frac{n-3}{(n-1)(n+3)} & \text{if } n \equiv 1\quad \mod 4, \\ 2^{n-1} \frac{n-5}{(n+2)(n-2)} & \text{if } n \equiv 2\quad \mod 4, \\ 2^{n-1} \frac{1}{n+1} & \text{if } n \equiv 3\quad \mod 4. \end{cases} \end{align*}$$

Proof Since $n \geq 5$ , $G_{\text {pr}}(\mathbb {F}_2^n)$ has at least four distinct eigenvalues, so Theorem 3.5 is applicable. First, we need to determine $\Delta = \max _{u \in V(G_{\text {pr}}(\mathbb {F}_2^n))} \{ (A^3)_{uu} \}$ . Since $G_{\text {pr}}(\mathbb {F}_2^n)$ is walk-regular, the diagonal entries of $A^3$ are all the same, so $\Delta = (A^3)_{\textbf {00}}$ . Now, $\Delta $ is exactly two times the number of triangles in the graph that vertex 0 is part of. Since $n \geq 5$ , vertex 0 can only be part of triangles where the vertices of the triangle differ in the same $F_i$ . However, since $q=2,$ there are no two distinct element in $\mathbb {F}_q^*$ , so vertex $\textbf {0}$ is not part of any triangles, and $\Delta =0$ .

We start with the case where n is even, or $n \equiv 0,2\ \mod 4$ . Then,

$$\begin{align*}\theta_0 = n+1, \quad \theta_r = 1-n, \end{align*}$$

and $\theta _s$ is the smallest eigenvalue $\geq -\frac {\theta _0^2+\theta _0\theta _r - \Delta }{\theta _0(\theta _r+1)}$ . Now,

$$\begin{align*}\frac{\theta_0^2+\theta_0\theta_r - \Delta}{\theta_0(\theta_r+1)} = \frac{(n+1)^2+(n+1)(1-n)}{(n+1)(2-n)} = \frac{2}{2-n}. \end{align*}$$

So $\theta _s:=2i-n-1$ is the smallest eigenvalue $\geq \frac {2}{n-2}$ . Then,

$$\begin{align*}2i-n-1 \geq \frac{2}{2-n} \Leftrightarrow 2i \geq n+1+\frac{2}{n-2} \Leftrightarrow i \geq \frac{n+1}{2} + \frac{1}{n-2}. \end{align*}$$

Since $n \geq 5$ , $\frac {1}{n-2} \leq \frac {1}{3} < \frac {1}{2}$ . Since i also has to be integral, we get $i \geq \frac {n+1}{2} + \frac {1}{2} = \frac {n}{2} +1$ . Since i has to be odd for $2i-n-1$ to be an eigenvalue when n is even, we get $i = \frac {n}{2}+1$ if $n \equiv 0\ \mod 4$ and $i = \frac {n}{2}+2$ if $n \equiv 2\ \mod 4$ . If $n \equiv 0\ \mod 4$ , we have $\theta _s = 1, \theta _{s+1} = -3$ . Then,

$$\begin{align*}\alpha_3(G_{\text{pr}}(\mathbb{F}_2^n)) \leq 2^n \frac{-(n+1)(1-3+1-n)-(-3)(1-n)}{(n+1-1)(n+1+3)(n+1-1+n)} = 2^{n-1} \frac{n^2-n+4}{n^2(n+4)}. \end{align*}$$

If $n \equiv 2\ \mod 4$ , we have $\theta _s = 3, \theta _{s+1} = -1$ . Then,

$$\begin{align*}\alpha_3(G_{\text{pr}}(\mathbb{F}_2^n)) \leq 2^n \frac{-(n+1)(3-1+1-n) - 3(-1)(1-n)}{(n+1-3)(n+1+1)(n+1-1+n)} = 2^{n-1} \frac{n-5}{(n+2)(n-2)}. \end{align*}$$

Next, we deal with the case, where n is odd, or $n \equiv 1,3\ \mod 4$ . Then,

$$\begin{align*}\theta_0 = n+1, \quad \theta_r = -n-1, \end{align*}$$

and $\theta _s$ is the smallest eigenvalue $\geq -\frac {\theta _0^2+\theta _0\theta _r - \Delta }{\theta _0(\theta _r+1)}$ . Now,

$$\begin{align*}\frac{\theta_0^2+\theta_0\theta_r - \Delta}{\theta_0(\theta_r+1)} = \frac{(n+1)^2+(n+1)(-n-1)}{(n+1)(-n)} = 0. \end{align*}$$

So $\theta _s:=2i-n-1$ is the smallest eigenvalue $\geq 0$ . Then,

$$\begin{align*}2i-n-1 \geq 0 \Leftrightarrow 2i \geq n+1 \Leftrightarrow i \geq \frac{n+1}{2}. \end{align*}$$

Since i has to be even for $2i-n-1$ to be an eigenvalue when n is odd, we get $i = \frac {n}{2}+1$ if $n \equiv 1\ \mod 4$ and $i = \frac {n+1}{2}$ if $n \equiv 3 \ \mod 4$ . If $n \equiv 1\ \mod 4$ , we have $\theta _s = 2, \theta _{s+1} = -2$ . Then,

$$\begin{align*}\alpha_3(G_{\text{pr}}(\mathbb{F}_2^n)) \leq 2^n \frac{-(n+1)(2-2-n-1)-2(-2)(-n-1)}{(n+1-2)(n+1+2)(n+1+n+1)} = 2^{n-1} \frac{n-3}{(n-1)(n+3)}. \end{align*}$$

If $n \equiv 3\ \mod 4$ , we have $\theta _s = 0, \theta _{s+1} = -4$ . Then,

$$ \begin{align*} &\alpha_3(G_{\text{pr}}(\mathbb{F}_2^n)) \leq 2^n \frac{-(n+1)(0-4-n-1)-0(-4)(-n-1)}{(n+1)(n+1+4)(n+1+n+1)} = \frac{2^{n-1}}{n+1}. \end{align*} $$

The upper bounds from Theorem 4.32 can be translated to upper bounds on $A^{\text {pr}}_2(n,4)$ via Lemma 3.1.

Corollary 4.33 The maximum cardinality of phase-rotation codes in $\mathbb {F}_2^n$ of minimum distance 4 with $n \geq 5$ is upper bounded by

(4.6) $$ \begin{align} A_2^{\text{pr}}(n,4) \leq \begin{cases} 2^{n-1} \frac{n^2-n+4}{n^2(n+4)} & \text{if } n \equiv 0\quad \mod 4, \\ 2^{n-1} \frac{n-3}{(n-1)(n+3)} & \text{if } n \equiv 1\quad \mod 4, \\ 2^{n-1} \frac{n-5}{(n+2)(n-2)} & \text{if } n \equiv 2\quad \mod 4, \\ 2^{n-1} \frac{1}{n+1} & \text{if } n \equiv 3\quad \mod 4. \end{cases} \end{align} $$

A comparison of these upper bounds from Corollary 4.33 to the Singleton-type bound from Theorem 4.22 follows next.

Proof The upper bound of the Singleton-type bound for $d=4$ and $q=2$ is $2^{n-3}$ if $4<1+\lceil n - \tfrac {n}{2} \rceil $ , which is exactly if $\tfrac {n}{2}> 3 \Leftrightarrow n >6$ . In this case, we compare the bounds and see when the bounds from Equation (4.6) are smaller than or equal to $2^{n-3}$ . For $n \equiv 0\ \mod 4,$ we have:

$$\begin{align*}2^{n-1} \frac{n^2-n+4}{n^2(n+4)} \leq 2^{n-3} \Leftrightarrow 2^2(n^2-n+4) \leq n^2(n+4) \Leftrightarrow n^3+4n \geq 16, \end{align*}$$

which is true since $n \geq 5$ . For $n \equiv 1\ \mod 4,$

$$\begin{align*}2^{n-1} \frac{n-3}{(n-1)(n+3)} &\leq 2^{n-3} \Leftrightarrow 2^2(n-3) \\ &\leq (n-1)(n+3) \Leftrightarrow n^2-2n+9 = (n-1)^2+8 \geq 0. \end{align*}$$

Also this is true. For $n \equiv 2\ \mod 4,$

$$\begin{align*}2^{n-1} \frac{n-5}{(n+2)(n-2)} &\leq 2^{n-3} \Leftrightarrow 2^2(n-5) \\ &\leq (n-2)(n+2) \Leftrightarrow n^2-4n+16 = (n-2)^2+12 \geq 0, \end{align*}$$

which is true. Lastly, for $n \equiv 3\ \mod 4,$ we get:

$$\begin{align*}2^{n-1} \frac{1}{n+1} \leq 2^{n-3} \Leftrightarrow 2^2 \leq n+1 \Leftrightarrow n \geq 3, \end{align*}$$

which is true by assumption on n.

In the cases that the Singleton-type bound equals $1$ , which is if $n=5,6$ , the bounds from Equation (4.6) give values of $2^4 \cdot \tfrac {2}{32} = 1$ and $2^5 \cdot \tfrac {1}{32} = 1$ for $n=5,6,$ respectively. Hence, the upper bounds on $A_2^{\text {pr}}(n,4)$ from Equation (4.6) are no worse than the Singleton-type upper bound from Theorem 4.22.

So the Ratio-type bound gives upper bounds on the size of phase-rotation codes that perform no worse than the Singleton-type bound for $k=3$ and $q=2$ . Now, we consider the Ratio-type bound for $k=3$ and $q \geq 3$ .

Theorem 4.35 Let $n \geq 3$ and $q \geq 3$ . Then,

$$\begin{align*}\alpha_3(G_{\text{pr}}({\mathbb{F}}_q^n)) \leq q^n \frac{n(n+2q-1)+q \lceil \tfrac{n-1}{q} \rceil \big(-2n-q+q \lceil \tfrac{n-1}{q} \rceil \big)}{q^3 \big( n+\lfloor \tfrac{1-n}{q} \rfloor \big) \big( n+1+ \lfloor \tfrac{1-n}{q} \rfloor \big)}. \end{align*}$$

Proof The eigenvalues of $G_{\text {pr}}({\mathbb {F}}_q^n)$ for $n\geq 3$ and $q \geq 3$ are $iq-n-1$ for $i=0,1, \ldots , n-1,n+1$ . Since $n\geq 3$ , there are at least four distinct eigenvalues. Since $G_{\text {pr}}({\mathbb {F}}_q^n)$ is regular, Theorem 3.5 is applicable. First, we need to determine $\Delta = \max _{u \in V(G_{\text {pr}}({\mathbb {F}}_q^n))} \{ (A^3)_{uu} \}$ . Similarly to the previous proof, $\Delta = (A^3)_{\textbf {00}}$ , which equals two times the number of triangles that vertex $\textbf {0}$ is part of. Again since $n \geq 3$ , vertex 0 is only part of triangles where the vertices of the triangle differ in the same $F_i$ . Then, 0 is part of $(n+1) {q-1 \choose 2}$ triangles since there are $n+1 F_i$ ’s and ${q-1 \choose 2}$ ways to choose two different elements in ${\mathbb {F}}_q^*$ . So $\Delta = 2(n+1){q-1 \choose 2} = (n+1)(q-1)(q-2)$ . Using the notation of Theorem 3.5, we have

$$\begin{align*}\theta_0=(n+1)(q-1), \quad \theta_r = -n-1, \end{align*}$$

and $\theta _s$ is the smallest eigenvalue $\geq - \frac {\theta _0^2+\theta _0\theta _r - \Delta }{\theta _0(\theta _r+1)}$ . Now,

$$\begin{align*}\frac{\theta_0^2+\theta_0\theta_r - \Delta}{\theta_0(\theta_r+1)} &= \frac{(n+1)^2(q-1)^2 + (n+1)(q-1)(-n-1) - (n+1)(q-1)(q-2)}{(n+1)(q-1) \cdot -n}\\ & = 2-q. \end{align*}$$

So $\theta _s:=iq-n-1$ is the smallest eigenvalue $\geq q-2$ . Then,

$$\begin{align*}iq-n-1 \geq q-2 \Leftrightarrow (i-1)q \geq n-1 \Leftrightarrow i \geq 1+ \frac{n-1}{q}. \end{align*}$$

Since i has to be equal to one of the integers $0,1, \ldots , n-1,n+1$ , take $i = 1+ \lceil \frac {n-1}{q} \rceil $ , which is a positive integer and at most $n-1$ . Then,

$$\begin{align*}\theta_s = (1+ \lceil \tfrac{n-1}{q} \rceil)q-n-1, \theta_{s+1} = \lceil \tfrac{n-1}{q} \rceil q-n-1. \end{align*}$$

Now, we obtain the following upper bound for $\alpha _3(G_{\text {pr}}({\mathbb {F}}_q^n))$ :

$$\begin{align*}q^n \Bigg( \frac{(n+1)(q-1)(q-2) - (n+1)(q-1) ((1+ \lceil \tfrac{n-1}{q} \rceil)q-n-1 + \lceil \tfrac{n-1}{q} \rceil q-n-1 -n-1) }{((n+1)(q-1) - ((1+ \lceil \tfrac{n-1}{q} \rceil)q-n-1))((n+1)(q-1) - ( \lceil \tfrac{n-1}{q} \rceil q-n-1))((n+1)(q-1) +n+1)} \end{align*}$$

$$\begin{align*}-\ \frac{((1+ \lceil \tfrac{n-1}{q} \rceil)q-n-1) ( \lceil \tfrac{n-1}{q} \rceil q-n-1)(-n-1)}{((n+1)(q-1) - ((1+ \lceil \tfrac{n-1}{q} \rceil)q-n-1))((n+1)(q-1) - ( \lceil \tfrac{n-1}{q} \rceil q-n-1))((n+1)(q-1) +n+1)} \Bigg) \end{align*}$$

$$\begin{align*}q^n \frac{n(n+2q-1)+q \lceil \tfrac{n-1}{q} \rceil \big(-2n-q+q \lceil \tfrac{n-1}{q} \rceil \big)}{q^3 \big( n+\lfloor \tfrac{1-n}{q} \rfloor \big) \big( n+1+ \lfloor \tfrac{1-n}{q} \rfloor \big)}.\\[-45pt] \end{align*}$$

The upper bound from Theorem 4.35 can be translated to upper bounds on $A_q^{\text {pr}}(n,4)$ via Lemma 3.1.

Corollary 4.36 The maximum cardinality of phase-rotation codes in ${\mathbb {F}}_q^n$ of minimum distance 4 with $n \geq 3$ and $q \geq 3$ is upper bounded by

(4.7) $$ \begin{align} A_q^{\text{pr}}(n,4) \leq q^n \frac{n(n+2q-1)+q \lceil \tfrac{n-1}{q} \rceil \big(-2n-q+q \lceil \tfrac{n-1}{q} \rceil \big)}{q^3 \big( n+\lfloor \tfrac{1-n}{q} \rfloor \big) \big( n+1+ \lfloor \tfrac{1-n}{q} \rfloor \big)}. \end{align} $$

We compare this upper bound from Corollary 4.36 with the Singleton-type upper bound from Theorem 4.22.

Proof The upper bound from the Singleton-type bound for $d=4$ is $q^{n-4+1}=q^{n-3}$ if $4<1+\lceil n - \tfrac {n}{q} \rceil $ . This is exactly when $n-\tfrac {n}{q}> 3 \Leftrightarrow n>3+\tfrac {3}{q-1}$ . If $q=3,4$ , then we need $n \geq 5$ , and if $q \geq 5$ , then $n \geq 4$ suffices. We consider these cases first. Define $m:= \lceil \tfrac {n-1}{q} \rceil $ . Then, $m-1 < \tfrac {n-1}{q} \leq m$ by definition of m. Since $\lfloor \tfrac {1-n}{q} \rfloor = -\lceil \tfrac {n-1}{q} \rceil = -m$ , the upper bound from Equation (4.7) becomes:

$$\begin{align*}q^n \frac{n(n+2q-1)+q m (-2n-q+q m)}{q^3(n-m)(n+1-m)}. \end{align*}$$

The latter is less than or equal to the Singleton-type upper bound if

$$\begin{align*}\frac{n(n+2q-1)+q m (-2n-q+q m)}{(n-m)(n+1-m)} \leq 1 \end{align*}$$

$$\begin{align*}\Leftrightarrow n(n+2q-1)+qm(-2n-q+qm) \leq (n-m)(n+1-m) \end{align*}$$

$$\begin{align*}\Leftrightarrow 2qn-n-2q m n -q^2m+q^2 m^2 \leq n -2mn -m +m^2. \end{align*}$$

Since $m,n,q$ are integral, the latter inequality can be shown to hold, using mathematical software, when $n \geq 3, q \geq 3$ , and $\smash {m-1 < \tfrac {n-1}{q} \leq m}$ . Note that by definition of m, we have $\smash {m-1 < \tfrac {n-1}{q} \leq m}$ . The conditions on n and q hold by the given assumptions on n and q.

Next, we consider the cases $q=3,4$ , $n=3,4$ and $q \geq 5$ , $n=3$ . Now, the Singleton-type bound gives an upper bound of $1$ . If $n=3$ , the upper bound from Equation (4.7) reduces to $1$ . However, for $n=4$ , $q=3,4$ , the upper bound of Equation (4.7) reduces to q, which is not less than or equal to $1$ . That finishes the proof.

So upper bounds on the size of phase-rotation codes obtained via the Ratio-type bound almost always perform no worse than the Singleton-type bound for $k=3$ and $q \geq 3$ .

We have shown theoretically that, for the phase-rotation metric, the Ratio-type bound performs no worse than the Singleton-type bound in most cases when the minimum distance is small, i.e., $d=2,3,4$ , and n is large enough. Next, we provide some computational results for larger values of the minimum distance. Consider all graphs $G_{\text {pr}}({\mathbb {F}}_q^n)$ with $n \geq 2$ , q a prime power and at most 1000 vertices, and consider $k=1,\ldots , \lceil \tfrac {q-1}{q}n \rceil -1$ . We compare the Inertia-type bound, the Ratio-type bound, and the Singleton-type bound in these instances. Note that the graph $G_{\text {pr}}({\mathbb {F}}_q^n)$ is not explicitly constructed, but its eigenvalues are calculated using Proposition 4.16. This implies that a comparison of the upper bounds to the actual value of the k-independence number is not possible. For all the considered instances where moreover $d<1+ \lceil n -\tfrac {n}{q} \rceil $ , the Ratio-type bound performs no worse than the Singleton-type bound. In some instances, like $n=5,6,q=3,k=3$ and $n=9,q=2,k=3,4$ , the Ratio-type bound improves on the Singleton-type upper bound. There are also some improvements with the Inertia-type bound compared to the Singleton-type. If $n=6,q=2,k=1$ or $n=8, q=2, k=1,3$ , the Inertia-type bound performs better than the Singleton-type bound and the Ratio-type bound.

In what follows, we show some more results for the Inertia-type bound and the Ratio-type bound. This time the graph $G_{\text {pr}}({\mathbb {F}}_q^n)$ is explicitly constructed, so the upper bounds on the k-independence number can be compared to the true k-independence number. The results for

$$\begin{align*}n=2,3,4, \quad q=2,3,4,5, \quad k=1,\ldots, \lceil \tfrac{q-1}{q}n \rceil-1, \end{align*}$$

$$\begin{align*}\text{and } n=5, \quad q=2,3, \quad k=1,\ldots, \lceil \tfrac{q-1}{q}n \rceil-1 \end{align*}$$

can be seen in Table 3. The columns “Inertia-type” and “Ratio-type” contain the value of the Inertia-type bound and the value of the Ratio-type bound, respectively, for the given graph instance. Similarly, the column “ $\vartheta (G^k)$ ” contains the value of the Lovász theta number and the column “Singleton-type” contains the value of the Singleton-type upper bound. Since it is computationally expensive to compute the Lovász theta number, the value is not computed for every graph instance. In that case, this is indicated by a dash in the corresponding entry of the table. The column “ $\alpha _k$ ” contains the value of the true k-independence number of that graph instance. Only the instances where the Inertia-type bound or the Ratio-type bound performed no worse than the Singleton-type bound are provided. A value in the columns “Inertia-type” and “Ratio-type” is indicated in bold when it is lower than the corresponding Singleton-type upper bound.

Table 3 Results of the Inertia-type bound and the Ratio-type bound for the phase-rotation metric, compared to the Singleton-type bound, the Lovász theta number $\vartheta (G^k)$ , and the actual k-independence number $\alpha _k$ . Improvements of the Inertia-type bound and the Ratio-type bound compared to the Singleton-type bound are in bold.

As expected from the theoretical results, the Ratio-type bound performs well. It performs at least as good as the Singleton-type bound in almost all tested instances, even in an instance with $k = 4$ , which was not included in the earlier theoretical analysis. Moreover, the Ratio-type bound improves on the Singleton-type bound in several instances and is also sharp in many instances. The performance of the Inertia-type bound, on the other hand, varies widely. In most instances, it performs worse than the Ratio-type bound. However, when $n=5, q=2, k=1,2,$ the Inertia-type bound performs equally good and is sharp. Moreover, when $n=4, q=2, k=1,$ the Inertia-type bound outperforms both the Ratio-type bound and the Singleton-type bound, and is equal to the k-independence number.

To sum up the results for the phase-rotation metric, we have seen that the spectral bounds improve the Singleton-type bound in several instances. For the Ratio-type bound, it was proven theoretically that in most instances, where the minimum distance is small (and n is large enough), the Ratio-type bound is at least as good as the Singleton-type bound. Some computational results also show improvement for the Ratio-type bound in several instances. For the Inertia-type bound, computational results show that there are a few instances where it outperforms the Ratio-type bound and the Singleton-type bound, while its overall performance varies widely.

5.1 Block metric

The block metric was introduced in [Reference Feng, Xu and Hickernell29].

Definition 5.1 Let $P=\{p_1, \ldots , p_m\}$ be a partition of $[n]$ . The block P-weight of $\textbf {x} \in {\mathbb {F}}_q^n$ is defined as

$$\begin{align*}w_P(\textbf{x}) := \min\left\{|I|: \text{supp}(\textbf{x}) \subseteq \bigcup_{i \in I} p_i\right\}. \end{align*}$$

The block P-distance between $\textbf {x},\textbf {y} \in {\mathbb {F}}_q^n$ is defined as $d_P(\textbf {x},\textbf {y}) := w_P(\textbf {x}-\textbf {y})$ .

Fix a partition $P = \{p_1, \dots , p_m\}$ of $[n]$ . Applying the Eigenvalue Method to the discrete metric space $({\mathbb {F}}_q^n, d_P)$ gives the block P-distance graph $G_P({\mathbb {F}}_q^n)$ . This graph satisfies condition (C1) and properties (P1) and (P2). Moreover, property 3 holds since $G_P({\mathbb {F}}_q^n)$ is not distance-regular in general. So both the Inertia-type bound and the Ratio-type bound, and their respective linear programs, can be applied to this graph.

The bounds obtained via the Eigenvalue Method can be compared to a Singleton-type bound: for a code $\mathcal {C} \subseteq {\mathbb {F}}_q^n$ of minimum block P-distance $d,$ it holds that

(5.1) $$ \begin{align} |\mathcal{C}| \leq q^{\sum_{j=d}^m \, p_j}, \end{align} $$

where w.l.o.g., $|p_1| \geq \cdots \geq |p_m|$ . This bound can easily be derived from the Singleton-type bound for the combinatorial metric, which can be found in [Reference Bossert and Sidorenko13], since the block metric is an example of a combinatorial metric. Now, we test the performance of the following instances:

$$\begin{align*}P=\big\{ \{1,2\},\{3\} \big\}, \big\{ \{1,2\},\{3,4\} \big\}, \big\{ \{1,2,3\},\{4,5\} \big\}, \quad q=2,3,4, \quad k=1,\ldots, m-1, \end{align*}$$

$$\begin{align*}\text{and } P=\big\{ \{1,2\},\{3,4\},\{5,6\} \big\}, \big\{ \{1,2,3\},\{4,5\},\{6\} \big\}, \quad q=2,3, \quad k=1,\ldots, m-1. \end{align*}$$

The results can be seen in Table 4. The columns “Inertia-type,” “Ratio-type,” and “Singleton-type” give the value of the Inertia-type bound, the Ratio-type bound, and the Singleton-type bound, respectively, for the given instance. The columns “ $\alpha _k$ ” and “ $\vartheta (G^k)$ ” contain the value of the k-independence number and the value of the Lovász theta number, respectively. For some instances, the Lovász theta number could not be calculated in reasonable time, which is indicated by a dash in the table. Since the Singleton-type bound always performs at least as good as the bounds obtained using the Eigenvalue Method, only the instances where either the Inertia-type bound or the Ratio-type bound attains the k-independence number are displayed in the table.

Table 4 Results of the Inertia-type bound and the Ratio-type for the block metric, compared to the Singleton-type bound, the Lovász theta number $\vartheta (G^k)$ , and the actual k-independence number $\alpha _k$ .

We see that the Ratio-type bound equals the k-independence number is some specific instances, while the Inertia-type bound is strictly larger in all tested instances. Notably, most instances where the Ratio-type bound is tight are of the form $|p_1|= \cdots = |p_m|$ .

5.2 Cyclic b-burst metric

The cyclic b-burst metric was introduced in [Reference Bridwell and Wolf14].

Definition 5.2 Let $2 \leq b \leq n-1$ . Define $A_i := \{i+j: j=1, \ldots , b\}$ for $i=0, \ldots , n-1$ , where the addition is done modulo n and $0\ \mod {n}$ is denoted as n. Let $\mathcal {A} := \{A_0, \ldots , A_{n-1}\}$ . The cyclic b-burst weight of $\textbf {x} \in {\mathbb {F}}_q^n$ is defined as

$$\begin{align*}w_b(\textbf{x}) := \min\left\{|I|: \text{supp}(\textbf{x}) \subseteq \bigcup_{i \in I} A_i\right\}. \end{align*}$$

The cyclic b-burst distance between $\textbf {x}, \textbf {y} \in {\mathbb {F}}_q^n$ is defined as $d_b(\textbf {x}, \textbf {y}) := w_b(\textbf {x} - \textbf {y})$ .

Example 5.1 To illustrate the set $\mathcal {A}$ from Definition 5.2, consider $n=5$ and $b=3$ . Then,

$$\begin{align*}A_0=\{1,2,3\}, \quad A_1=\{2,3,4\}, \quad A_2=\{3,4,5\}, \quad A_3=\{4,5,1\}, \quad A_4=\{5,1,2\}. \end{align*}$$

Fix $2 \leq b \leq n-1$ . Applying the Eigenvalue Method to the discrete metric space $({\mathbb {F}}_q^n,d_b)$ gives the cyclic b-burst distance graph $G_b({\mathbb {F}}_q^n)$ . This graph satisfies condition (C1) and properties (P1) and (P2). Moreover, $G_b({\mathbb {F}}_q^n)$ is not distance-regular in general, so property 3 holds. This means both the Inertia-type bound and the Ratio-type bound, and their respective linear programs, can be applied to this graph.

The bounds obtained via the Eigenvalue Method can be compared to a Singleton-type bound: for a code $\mathcal {C} \subseteq {\mathbb {F}}_q^n$ of minimum cyclic b-burst distance $d,$ it holds that

(5.2) $$ \begin{align} |\mathcal{C}| \leq q^{n-b(d-1)}. \end{align} $$

This bound can be derived from the Singleton-type bound for the combinatorial metric in [Reference Bossert and Sidorenko13], since the cyclic b-burst metric is an example of a combinatorial metric. This bound is also known as the extended Reiger bound (see [Reference Villalba, Orozco and Blaum55]). We test the performance of the spectral bounds in the following instances:

$$\begin{align*}n=3,4,5, \quad q=2,3, \quad b=2,\ldots, n-1, \quad k=1, \ldots \lceil \tfrac{n}{b} \rceil -1, \end{align*}$$

$$\begin{align*}\text{and } n=3,4, \quad q=5, \quad b=2, \quad k=1. \end{align*}$$

The results can be seen in Table 5. The columns “Inertia-type,” “Ratio-type,” and “Singleton-type” give the value of the Inertia-type bound, the Ratio-type bound, and the Singleton-type bound, respectively, for the given instance. The columns “ $\alpha _k$ ” and “ $\vartheta (G^k)$ ” contain the value of the k-independence number and the value of the Lovász theta number, respectively. Since the Singleton-type bound always performs at least as good as the bounds obtained using the Eigenvalue Method, only the instances where either the Inertia-type bound or the Ratio-type bound attains the k-independence number are displayed in the table.

Table 5 Results of the Inertia-type bound and the Ratio-type for the cyclic b-burst metric, compared to the Singleton-type bound, the Lovász theta number $\vartheta (G^k)$ , and the actual k-independence number $\alpha _k$ .

Table 5 shows that the Inertia-type bound is not tight in any tested instances, while the Ratio-type bound is tight in some instances. However, it is not immediately clear why those instances give a tightness for the Ratio-type bound.

5.3 Varshamov metric

The Varshamov metric, also known as the asymmetric metric, was introduced in [Reference Varshamov54].

Definition 5.3 The Varshamov distance between $\textbf {x},\textbf {y} \in \mathbb {F}_2^n$ is defined as

$$\begin{align*}d_{\text{Var}}(\textbf{x},\textbf{y}) := \frac{1}{2} \left( w_{\text{H}}(\textbf{x}-\textbf{y}) + \left|w_{\text{H}}(\textbf{x})- w_{\text{H}}(\textbf{y})\right| \right), \end{align*}$$

where $w_{\text {H}}$ denotes the Hamming weight.

Another definition of the Varshamov distance between $\textbf {x}=(x_1,\ldots ,x_n), \textbf {y}=(y_1,\ldots ,y_n) \in \mathbb {F}_2^n$ is given in [Reference Rao and Chawla45]:

$$\begin{align*}d_{\text{Var}}(\textbf{x},\textbf{y}) := \max \{N_{01}(\textbf{x},\textbf{y}), N_{10}(\textbf{x},\textbf{y})\}, \end{align*}$$

where

$$\begin{align*}N_{01}(\textbf{x},\textbf{y}) := \left|\{i: x_i = 0, y_i = 1\} \right|, \qquad N_{10}(\textbf{x},\textbf{y}) := \left|\{i: x_i = 1, y_i = 0\}\right|. \end{align*}$$

In [Reference Kløve39, Lemma 2.1] the equivalence of both definitions is proven.

Applying the Eigenvalue Method to the discrete metric space $(\mathbb {F}_2^n, d_{\text {Var}})$ gives the Varshamov distance graph $G_{\text {Var}}(\mathbb {F}_2^n)$ . This graph satisfies condition (C1) and property (P3). However, desired properties (P1) and (P2) do not hold. This means only the Inertia-type bound can be applied to this graph.

The bound obtained via the Eigenvalue Method can be compared to a Plotkin-type bound [Reference Borden12] and to a bound due to Varshamov [Reference Varshamov53]. The latter bound states that for a code $\mathcal {C} \subseteq \mathbb {F}_2^n$ of minimum Varshamov distance $d,$ it holds that

(5.3) $$ \begin{align} |\mathcal{C}| \leq \frac{2^{n+1}}{\sum\limits_{i=0}^{d-1} {\lfloor n/2 \rfloor \choose i} + {\lceil n/2\rceil \choose i}}. \end{align} $$

Note that an integer programming bound also exists for codes in the Varshamov metric (see, e.g., [Reference Delsarte and Piret25]). We test some instances of the graph, specifically,

$$\begin{align*}n=2,\ldots, 8, \quad k=1,\ldots n-1 \text{ except } (n,k)=(8,7). \end{align*}$$

The results can be seen in Table 6. The columns “Inertia-type,” “Plotkin-type,” and “Varshamov” give the value of the Inertia-type bound, the Plotkin-type bound, and the bound due to Varshamov, respectively, for the given instance. The columns “ $\alpha _k$ ” and “ $\vartheta (G^k)$ ” contain the value of the k-independence number and the value of the Lovász theta number, respectively. Since either the Plotkin-type bound or the bound due to Varshamov always performs at least as good as the Inertia-type bound, only the instances where the Inertia-type bound attains the k-independence number are displayed in the table.

Table 6 Results of the Inertia-type bound for the Varshamov metric, compared to the Plotkin-type bound, the bound from Varshamov, the Lovász theta number $\vartheta (G^k)$ , and the actual k-independence number $\alpha _k$ .

We can see in Table 6 that the instances where the Inertia-type bound is tight are those where k is close to n. In all these instances, the k-independence number equals 2.