On automorphism groups of binary cyclic codes

TL;DR

This paper investigates the automorphism groups of binary cyclic codes, providing new constructions for long-length codes that aid in understanding their algebraic symmetries, which are crucial for decoding and applications.

Contribution

The paper introduces constructions for binary cyclic codes of long lengths that enable the determination of their full automorphism groups, advancing algebraic analysis of these codes.

Findings

Constructed binary cyclic codes with long lengths.

Facilitated determination of full automorphism groups.

Enhanced understanding of algebraic symmetries in cyclic codes.

Abstract

A lot of attention has been paid to the investigation of the algebraic properties of linear codes. In most cases, this investigation involves the determination of required code automorphisms, which are useful for decoders, such as the automorphism ensemble decoder. It is worth noting that the examination of the automorphism groups of discrete symmetric objects has long been a highly regarded field of research. Cyclic codes, as a significant subclass of linear codes, can be constructed and analyzed using algebraic methods. And due to its cyclic nature, they have efficient encoding and decoding algorithms. To date, cyclic codes have found applications in various domains, including consumer electronics, data storage systems, and communication systems. In this paper, we investigate the full automorphism groups of binary cyclic codes. In particular, we present constructions for binary cyclic codes of long lengths that facilitate the determination of the full automorphism groups.