Long quasi-polycyclic $t-$CIS codes

TL;DR

This paper investigates the structure and enumeration of long quasi-polycyclic $t$-CIS codes, establishing their asymptotic existence and demonstrating infinite families with favorable distance properties.

Contribution

It introduces and analyzes quasi-polycyclic $t$-CIS codes, extending previous work on quasi-cyclic and quasi-twisted codes, and provides asymptotic existence results under certain conjectures.

Findings

Infinite families of long QC and QT $t$-CIS codes with good relative distance.

Asymptotic existence results for one-generator $t$-CIS codes.

Extension of results to the newly introduced quasi-polycyclic codes.

Abstract

We study complementary information set codes of length $tn$ and dimension $n$ of order $t$ called ($t-$CIS code for short). Quasi-cyclic and quasi-twisted $t$-CIS codes are enumerated by using their concatenated structure. Asymptotic existence results are derived for one-generator and have co-index $n$ by Artin's conjecture for quasi cyclic and special case for quasi twisted. This shows that there are infinite families of long QC and QT $t$-CIS codes with relative distance satisfying a modified Varshamov-Gilbert bound for rate $1/t$ codes. Similar results are defined for the new and more general class of quasi-polycyclic codes introduced recently by Berger and Amrani.