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

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 U n , q of n -th roots-of-unity in the splitting field F q m of x n − 1 over F q is fixed set-wise by an F q -linear map on F q m that is not of the form x ⟶ α x q j . − There is a non-cyclic linear recurring sequence s of period n (so with s not of the form s i = α ξ i for i ≥ 0) with associated characteristic polynomial being irreducible over F q , such that U n , q = { s 0 , ... , s n − 1 }. In that case, the group U n , 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]. [ABSTRACT FROM AUTHOR]