An Updated Database of $$\mathbb {Z}_4$$ Codes and an Open Problem About Quasi-cyclic Codes

Abstract

Research on codes over finite rings has intensified after the discovery that some of the best binary nonlinear codes can be obtained as images of $$\mathbb {Z}_4$$ -linear codes. Codes over various finite rings have been a subject of much research in coding theory after this discovery. Many of these rings are extensions of $$\mathbb {Z}_4$$ and numerous new linear codes over $$\mathbb {Z}_4$$ have been found in the last decade. Due to the special place of $$\mathbb {Z}_4$$ , an online database of $$\mathbb {Z}_4$$ codes was created in 2007. The original database on $$\mathbb {Z}_4$$ codes recently became unavailable. The purpose of this paper is to introduce a new and updated database of $$\mathbb {Z}_4$$ codes. We have made major updates to the original database by adding 8699 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 now added a large number of non-free codes. In fact, of the 8699 codes we have added, 7631 of them are non-free. Our database will be further updated by incorporating data from [16]. We also state an open problem of great theoretical interest that arose from computational observations.