Basic properties of m-sequences

Let R be a finite commutative ring and let a be a periodic sequence of elements in R. Let T be the period of a.

Definition 10.1.1 The sequence a is an m-sequence (over the ring R) of rank r (or degree r or span r) if it can be generated by a linear feedback shift register with r cells, and if every nonzero block of length r occurs exactly once in each period of a.

In other words, the sequence a is the output sequence of an LFSR that cycles through all possible nonzero states before it repeats. The second condition in the definition also says that a is a punctured de Bruijn sequence. See Section 8.2.4. In this section we recall standard results about m-sequences which have been known since the early 1900s [41] and which may be found in Golomb's book in the binary case [61, 62].

Proposition 10.1.2Suppose a is an m-sequence of rank r over a (finite commutative) ring R, generated by an LFSR with connection polynomial q(x) ∈ R[x] of degree r. Then the following hold.

1. 1. The ring R is a field and the connection polynomial q(x) is a primitive polynomial.

2. […]