New record-breaking binary linear codes constructed from group codes

TL;DR

The paper addresses advancing binary linear codes by combining group-ring constructions with automorphism-based refinements. It introduces $G$-matrix codes $C(v)$ from semidirect-product groups $G$ and derives explicit block-circulant representations, enabling structured parameter optimization. Through automorphism-decomposition (Construction X) and computational search in Magma, the authors obtain 77 new record-breaking binary codes, including notable instances such as $[108,23,36]$ and $[120,31,34]$, with extensive data made publicly available. The work demonstrates a systematic algebraic framework to push code bounds and suggests extending the methodology to non-binary codes and longer lengths.

Abstract

In this paper, we employ group rings and automorphism groups of binary linear codes to construct new record-breaking binary linear codes. We consider the semidirect product of abelian groups and cyclic groups and use these groups to construct linear codes. Finally, we obtain some linear codes which have better parameters than the code in \cite{bib5}. All the calculation results and corresponding data are listed in the paper or posted online.

Theorems & Definitions (18)

• Lemma 1.1
• Lemma 2.1
• Lemma 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 4.1
• Theorem 4.2
• proof