On the algebraic representation of selected optimal non-linear binary codes

TL;DR

This paper explores an algebraic framework for representing optimal non-linear binary codes over Z4, enabling new insights, simplified decoding algorithms, and potential code constructions through Fourier analysis and subgroup structures.

Contribution

It introduces a novel algebraic representation of certain optimal non-linear binary codes using subgroups of units in a ring, facilitating decoding and analysis.

Findings

New algebraic representation of Best's (10, 40, 4) code as a subgroup coset in Z4[Z5]

Simplified algebraic decoding algorithm for Best's code

Application of techniques to analyze Julin's (12, 144, 4) code and the (12, 24, 12) Hadamard code

Abstract

Revisiting an approach by Conway and Sloane we investigate a collection of optimal non-linear binary codes and represent them as (non-linear) codes over Z4. The Fourier transform will be used in order to analyze these codes, which leads to a new algebraic representation involving subgroups of the group of units in a certain ring. One of our results is a new representation of Best's (10, 40, 4) code as a coset of a subgroup in the group of invertible elements of the group ring Z4[Z5]. This yields a particularly simple algebraic decoding algorithm for this code. The technique at hand is further applied to analyze Julin's (12, 144, 4) code and the (12, 24, 12) Hadamard code. It can also be used in order to construct a (non-optimal) binary (14, 56, 6) code.