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

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.

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