Automorphism groups of cyclic codes

TL;DR

This paper investigates the automorphism groups of binary cyclic codes, providing explicit constructions for groups structured as direct products or wreath products of symmetric groups, with applications in lattice graphs and permutation decoding.

Contribution

It introduces new explicit constructions of cyclic codes with automorphism groups as complex group products, expanding understanding of their symmetry properties.

Findings

Automorphism groups can be expressed as direct products of symmetric groups.

Automorphism groups can be represented as iterated wreath products of symmetric groups.

Some codes relate to regular lattice graphs and permutation decoding techniques.

Abstract

In this article we study the automorphism groups of binary cyclic codes. In particular, we provide explicit constructions for codes whose automorphism groups can be described as (a) direct products of two symmetric groups or (b) iterated wreath products of several symmetric groups. Interestingly, some of the codes we consider also arise in the context of regular lattice graphs and permutation decoding.