The permutation automorphism groups of irreducible cyclic codes

Abstract

The study of permutation automorphism groups of cyclic codes is a central topic in algebraic coding theory. A cyclic code over $\mathbb{F}_q$ is called irreducible if its check polynomial is irreducible over $\mathbb{F}_q$. Such a code is standard if its permutation automorphism group is equal to the group generated by the cyclic shift and the Frobenius automorphism, and non-standard otherwise. In this paper, we give a complete classification of all non-standard non-degenerate irreducible cyclic codes, using the classification of finite simple groups. Our result shows that, apart from a small number of explicit exceptional families and their descendants under certain secondary constructions, every non-degenerate irreducible cyclic code is standard, and up to four explicit exceptions, every degenerate cyclic code is non-standard. This classification has several consequences. First, it yields a general description of non-standard linear recurring sequence subgroups, extending the earlier work of Brison and Nogueira; secondly it establishes the Schmidt-White conjecture for all non-standard irreducible cyclic codes. Moreover, our results provide strong evidence in support of the conjecture of Berger and Charpin that almost all cyclic codes are standard.