Codes from Zero-divisors and Units in Group Rings

TL;DR

This paper introduces a novel method for constructing codes from group rings by leveraging their matrix representations, enabling the creation of diverse codes with specific properties like LDPC or self-duality.

Contribution

It presents a new approach focusing on encodings from group rings rather than ideals, broadening the types of codes that can be constructed using algebraic structures.

Findings

Codes can be constructed from units and zero-divisors in group rings.

Matrix algebra facilitates the creation of generator and check matrices for these codes.

The method allows for codes over various rings, including integers and matrix rings.

Abstract

We describe and present a new construction method for codes using encodings from group rings. They consist primarily of two types: zero-divisor and unit-derived codes. Previous codes from group rings focused on ideals; for example cyclic codes are ideals in the group ring over a cyclic group. The fresh focus is on the encodings themselves, which only under very limited conditions result in ideals. We use the result that a group ring is isomorphic to a certain well-defined ring of matrices, and thus every group ring element has an associated matrix. This allows matrix algebra to be used as needed in the study and production of codes, enabling the creation of standard generator and check matrices. Group rings are a fruitful source of units and zero-divisors from which new codes result. Many code properties, such as being LDPC or self-dual, may be expressed as properties within the group ring thus enabling the construction of codes with these properties. The methods are general enabling the construction of codes with many types of group rings. There is no restriction on the ring and thus codes over the integers, over matrix rings or even over group rings themselves are possible and fruitful.