Non-standard linear recurring sequence subgroups and automorphisms of irreducible cyclic codes

TL;DR

This paper characterizes when irreducible cyclic codes have extra automorphisms, linking it to non-standard linear recurring sequence subgroups, and provides classifications and constructions for such codes.

Contribution

It establishes a precise connection between non-standard linear recurring sequence subgroups and non-standard automorphisms of irreducible cyclic codes, including a complete classification for dimension two.

Findings

Extra automorphisms occur precisely when the pair (n, q) is non-standard.

Constructs families of NSIC-codes using lifting and extension methods.

Classifies all NSIC-codes of dimension two.

Abstract

Let \(\cU\) be the multiplicative group of order~\(n\) in the splitting field \(\bbF_{q^m}\) of \(x^n-1\) over the finite field \(\bbF_q\). Any map of the form \(x\rightarrow cx^t\) with \(c\in \cU\) and \(t=q^i\), \(0\leq i<m\), is \(\bbF_q\)-linear on~\(\bbF_{q^m}\) and fixes \(\cU\) set-wise; maps of this type will be called {\em standard\/}. Occasionally there are other, {\em non-standard\/} \(\bbF_q\)-linear maps on~\(\bbF_{q^m}\) fixing \(\cU\) set-wise, and in that case we say that the pair \((n, q)\) is {\em non-standard\/}. We show that an irreducible cyclic code of length~\(n\) over \(\bbF_q\) has ``extra'' permutation automorphisms (others than the {\em standard\/} permutations generated by the cyclic shift and the Frobenius mapping that every such code has) precisely when the pair \((n, q)\) is non-standard; we refer to such irreducible cyclic codes as {\em non-standard\/} or {\em NSIC-codes\/}. In addition, we relate these concepts to that of a non-standard linear recurring sequence subgroup as investigated in a sequence of papers by Brison and Nogueira. We present several families of NSIC-codes, and two constructions called ``lifting'' and ``extension'' to create new NSIC-codes from existing ones. We show that all NSIC-codes of dimension two can be obtained in this way, thus completing the classification for this case started by Brison and Nogueira.