Projection decoding of some binary optimal linear codes of lengths 36 and 40

TL;DR

This paper introduces an efficient projection decoding method for certain binary optimal linear codes of lengths 36 and 40, enabling error correction up to weight 3, which was previously unknown for these codes.

Contribution

The authors demonstrate that specific optimal binary linear codes of lengths 36 and 40 can be decoded efficiently using projection onto shorter codes over GF(4), a novel approach for these codes.

Findings

Decoding algorithms correct errors of weight up to 3.

Codes have more codewords than comparable self-dual codes.

Projection decoding applies to previously undecodable optimal codes.

Abstract

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes such as cyclic codes, Reed-Solomon codes, and Reed-Muller codes have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, it is a natural question whether which optimal linear codes have an efficient decoding. We show that two binary optimal $[36,19,8]$ linear codes and two binary optimal $[40,22,8]$ codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal $[36,19,8]$ and $[40,22,8]$ codes. We project them onto the much shorter length linear $[9,5,4]$ and $[10, 6, 4]$ codes over $GF(4)$, respectively. This decoding algorithms, called {\em projection decoding}, can correct errors of weight up to 3. These $[36,19,8]$ and $[40,22,8]$ codes respectively have more codewords than any optimal self-dual $[36, 18, 8]$ and $[40,20,8]$ codes for given length and minimum weight, implying that these codes more practical.