Relations between the Local Weight Distributions of a Linear Block Code, Its Extended Code, and Its Even Weight Subcode

TL;DR

This paper explores the relationships between local weight distributions of binary linear codes, their extended versions, and even weight subcodes, providing methods to derive distributions for specific codes and improving existing algorithms.

Contribution

It establishes new relations between local weight distributions of codes and their extensions, and enhances algorithms for computing these distributions.

Findings

Derived local weight distributions for specific BCH and Reed-Muller codes.

Established relations allowing computation of distributions from extended codes.

Improved the algorithm for calculating local weight distributions.

Abstract

Relations between the local weight distributions of a binary linear code, its extended code, and its even weight subcode are presented. In particular, for a code of which the extended code is transitive invariant and contains only codewords with weight multiples of four, the local weight distribution can be obtained from that of the extended code. Using the relations, the local weight distributions of the $(127,k)$ primitive BCH codes for $k\leq50$, the $(127,64)$ punctured third-order Reed-Muller, and their even weight subcodes are obtained from the local weight distribution of the $(128,k)$ extended primitive BCH codes for $k\leq50$ and the $(128,64)$ third-order Reed-Muller code. We also show an approach to improve an algorithm for computing the local weight distribution proposed before.