An Updated Database of $\mathbb{Z}_4$ Codes

TL;DR

This paper introduces an extensively updated online database of $\\mathbb{Z}_4$ codes, adding 8701 new codes found through exhaustive searches and algorithm modifications, significantly expanding resources for coding theory research.

Contribution

The paper presents a major update to the $\\mathbb{Z}_4$ codes database, including new codes discovered via enhanced algorithms and exhaustive searches, especially non-free codes.

Findings

Added 8701 new $\\mathbb{Z}_4$ codes to the database.

Major updates include non-free codes, increasing diversity.

Modified ASR algorithm for $\\mathbb{Z}_4$ to find new codes.

Abstract

Research on codes over finite rings has intensified since the discovery in 1994 of the fact that some best binary non-linear codes can be obtained as images of $\mathbb{Z}_4$-linear codes. Codes over many different finite rings has been a subject of much research in coding theory after this discovery. Many of these rings are extensions of $\mathbb{Z}_4$. As a result, an online database of $\mathbb{Z}_4$ was created in 2008. The URL of the original database on $\mathbb{Z}_4$ codes has recently changed. The purpose of this paper is to introduce the new, updated database of $\mathbb{Z}_4$ codes. We have made major updates to the database by adding 8701 new linear codes over $\mathbb{Z}_4$. These codes have been found through exhaustive computer searches on cyclic codes and by an implementation of the ASR search algorithm that has been remarkably fruitful to obtain new linear codes from the class of quasi-cyclic (QC) and quasi-twisted (QT) codes over finite fields. We made modifications to the ASR algorithm to make it work over $\mathbb{Z}_4$. The initial database contained few codes that were not free. We have added a large number of non-free codes. In fact, of the 8701 codes we have added, 7631 of them are non-free.