Computing the distance distribution of systematic non-linear codes

TL;DR

This paper introduces a Groebner basis method for efficiently computing the weight and distance distributions of systematic non-linear codes, surpassing brute-force checks and enabling analysis of code families.

Contribution

The paper presents a novel Groebner basis approach for calculating distributions of systematic non-linear codes, extending beyond brute-force methods and applicable to code families.

Findings

Method computes weight and distance distributions efficiently.

Outputs all closest pairs in the code.

Can be extended to code families.

Abstract

The most important families of non-linear codes are systematic. A brute-force check is the only known method to compute their weight distribution and distance distribution. On the other hand, it outputs also all closest word pairs in the code. In the black-box complexity model, the check is optimal among closest-pair algorithms. In this paper we provide a Groebner basis technique to compute the weight/distance distribution of any systematic non-linear code. Also our technique outputs all closest pairs. Unlike the check, our method can be extended to work on code families.