Conditions for preserving code structure in quaternary quasi cyclic codes
Resolution of an Open Problem on Quasi-Cyclic Codes over $\mathbb{Z}_4$ and New Quaternary Linear Codes
Information Theory
Summary
Some special codes used in error correction are built using smaller building blocks called cyclic codes over a type of number system called Z4. It was unclear when certain structural properties of these building blocks carry over to bigger codes made from them. The authors solved this problem by finding clear rules that tell when the structure stays the same. They also used their findings to discover new codes that have better error correction properties than known ones.
What this means in practice
- •For communications engineers: Design quasi-cyclic codes over Z4 with guaranteed structure preservation to improve error correction performance in data transmission.
- •For coding theory software developers: Create tools that automate checking code type preservation in constructing quaternary quasi-cyclic codes, improving code search and validation.
Authors
Nuh Aydin, Aditya Tyagi
Abstract
Given a cyclic code $C_g = \langle g(x) \rangle$ of odd length $m$ over $\mathbb{Z}_4$, one common way to build a quasi-cyclic (QC) code is to pick $f_1, \dots, f_\ell \in \mathbb{Z}_4[x]$ and let $C = \langle (f_1 g, \dots, f_\ell g) \rangle$. Because $\mathbb{Z}_4$ is not a field, the type of $C_g$ ($4^{k_1}2^{k_2}$) is not necessarily inherited by $C$. Determining conditions under which the type of $C_g$ is inherited by $C$ was posed as an open problem recently in \cite{AydinLuOnta2023}. In this paper, we settle this problem. We first give two sufficient conditions for the type to be preserved: one requires a single $f_i$ to be coprime to $x^m-1$ over $\mathbb{F}_2$, the other only requires the $f_i$ to be jointly coprime to it. Neither condition is necessary in general. Using the fact that $x^m-1$ is squarefree for odd $m$, we decompose $\mathbb{Z}_4[x]/\langle x^m-1\rangle$ into a product of finite chain rings using Chinese remainder theorem (CRT) and derive a condition on $f_1, \dots, f_\ell$ that is both necessary and sufficient for $C$ to match the type of $C_g$. This condition depends only on the irreducible factors of $x^m-1$ where $g$ does not already vanish. This also yields a simple test for when $C$ is a free $\mathbb{Z}_4$-module. Finally, we report many new QC codes over $\mathbb{Z}_4$, found by computer searches using Magma software~\cite{Magma1997} guided by this criterion, with Lee distances greater than previously known codes of the same type.