Non-standard linear recurring sequence subgroups and automorphisms of irreducible cyclic codes (2202.07917v1)

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 calledlifting'' 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.