A note on reducing the computation time for minimum distance and equivalence check of binary linear codes

TL;DR

This paper proposes methods using Gray codes and combinatorial designs to improve the efficiency of computing minimum distances and checking code equivalence in binary linear codes, especially under memory constraints.

Contribution

It introduces a novel approach employing Gray codes with constant weight words and combinatorial designs to reduce computation time in code analysis.

Findings

Gray code method significantly speeds up minimum distance computation.

Combinatorial designs help in code equivalence testing under memory limitations.

The approaches are practical for large code sets with limited computational resources.

Abstract

In this paper we show the usability of the Gray code with constant weight words for computing linear combinations of codewords. This can lead to a big improvement of the computation time for finding the minimum distance of a code. We have also considered the usefulness of combinatorial $2$-$(t,k,1)$ designs when there are memory limitations to the number of objects (linear codes in particular) that can be tested for equivalence.