The main aim of this chapter will be to prove the MDS conjecture, which is a conjecture relating to maximum distance separable (MDS) codes. The conjecture can be stated without reference to MDS codes and was first proposed, or at least considered, by Beniamino Segre in the 1950s when coding theory was still in its inception. We will not state the full conjecture to begin with, but as a motivation we state a direct consequence of the proof of the conjecture over prime fields.

Theorem 7.1Let p be a prime and k be a positive integer, such that 2 ≤ k ≤ p. A k × (p + 2) integer matrix has a k × k submatrix whose determinant is zero modulo p.

We shall prove a lot more than Theorem 7.1, but for the moment we just note that it is optimal in two ways. If k = p + 1 then it is not true that a k × (p + 2) integer matrix has a k × k submatrix whose determinant is zero modulo p. For example, if we extend the (p + 1) × (p + 1) identity matrix with a column of all ones, then the resulting (p + 1) × (p + 2) matrix is a matrix all of whose (p + 1) × (p + 1) submatrices have determinant ± 1. It is also not true that if k ≤ p then a k × (p + 1) integer matrix must have a k × k submatrix whose determinant is zero modulo p. We can construct a k × (p + 1) matrix from Example 7.4, all of whose k × k submatrices are not zero modulo p.

Singleton bound

The bound in the following theorem is called the Singleton bound.

Theorem 7.2A block code C ⊆ An with minimum distance d satisfies

Proof Consider any n − (d − 1) coordinates.