Constructions of Self-Dual and Formally Self-Dual Codes from Group Rings

TL;DR

This paper presents improved methods for constructing self-dual and formally self-dual codes from group rings over finite commutative Frobenius rings, revealing new insights into their structure and limitations.

Contribution

It refines existing constructions by removing unnecessary conditions, connects codes to ideals in group rings, and identifies group restrictions for extremal codes.

Findings

Improved construction conditions for self-dual codes.

Codes correspond to ideals in group rings with specific automorphism groups.

Certain extremal codes cannot be produced by common construction techniques.

Abstract

We give constructions of self-dual and formally self-dual codes from group rings where the ring is a finite commutative Frobenius ring. We improve the existing construction given in \cite{Hurley1} by showing that one of the conditions given in the theorem is unnecessary and moreover it restricts the number of self-dual codes obtained by the construction. We show that several of the standard constructions of self-dual codes are found within our general framework. We prove that our constructed codes correspond to ideals in the group ring $RG$ and as such must have an automorphism group that contains $G$ as a subgroup. We also prove that a common construction technique for producing self-dual codes cannot produce the putative $[72,36,16]$ Type~II code. Additionally, we show precisely which groups can be used to construct the extremal Type II codes over length 24 and 48.