Non-standard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes I

Abstract

A pair (n,q) with q a prime power and n a positive integer for which gcd⁡(n,q)=1 is non-standard if one of the following properties holds, which we will show to be equivalent.− The group Un,q of n-th roots-of-unity in the splitting field Fqm of xn−1 over Fq is fixed set-wise by an Fq-linear map on Fqm that is not of the form x⟶αxqj.− There is a non-cyclic linear recurring sequence s of period n (so with s not of the form si=αξi for i≥0) with associated characteristic polynomial being irreducible over Fq, such that Un,q={s0,…,sn−1}. In that case, the group Un,q is called non-standard by Brison and Nogueira, who studied this phenomenon in a sequence of papers.− A q-ary irreducible cyclic code of length n has “extra” permutation automorphisms (some well-known examples are the (duals of the) Golay codes and binary simplex codes for n≥7). We will refer to such codes as non-standard irreducible cyclic codes or NSIC-codes.We first investigate non-standard linear recurring sequence subgroups and establish the equivalence between the first and second of the above properties. Then we investigate permutation automorphisms of general (repeated-root) cyclic codes and irreducible cyclic codes and establish the equivalence between the first and third of these properties.We also introduce a notion of non-standardness for general cyclic codes, and relate it to our earlier definition of NSIC-codes.This paper is the first part of an expanded version of the arXiv paper [28].