Computing coset leaders of binary codes

TL;DR

This paper introduces an efficient algorithm for computing all coset leaders of binary codes, leveraging techniques related to Gr"obner representations to improve over exhaustive search methods.

Contribution

It presents a novel algorithm that computes the set of coset leaders and provides a Gr"obner representation of binary codes with linear complexity relative to the number of coset leaders.

Findings

Algorithm efficiently computes coset leaders

Provides Gr"obner representation of binary codes

Complexity is linear in the number of coset leaders

Abstract

We present an algorithm for computing the set of all coset leaders of a binary code $\mathcal C \subset \mathbb{F}_2^n$. The method is adapted from some of the techniques related to the computation of Gr\"obner representations associated with codes. The algorithm provides a Gr\"obner representation of the binary code and the set of coset leaders $\mathrm{CL}(\mathcal C)$. Its efficiency stands of the fact that its complexity is linear on the number of elements of $\mathrm{CL}(\mathcal C)$, which is smaller than exhaustive search in $\mathbb{F}_2^n$.