On cyclotomic cosets and code constructions

TL;DR

This paper explores properties of q-ary cyclotomic cosets modulo n=q^m-1, leading to new classical cyclic codes, quantum CSS codes, and convolutional codes with improved parameters.

Contribution

It introduces novel properties of cyclotomic cosets that enable the construction of better-performing quantum and convolutional codes.

Findings

New bounds for code parameters are established.

Constructed CSS codes outperform existing ones.

Constructed convolutional codes have higher free distance.

Abstract

New properties of $q$-ary cyclotomic cosets modulo $n = q^{m} - 1$, where $q \geq 3$ is a prime power, are investigated in this paper. Based on these properties, the dimension as well as bounds for the designed distance of some families of classical cyclic codes can be computed. As an application, new families of nonbinary Calderbank-Shor-Steane (CSS) quantum codes as well as new families of convolutional codes are constructed in this work. These new CSS codes have parameters better than the ones available in the literature. The convolutional codes constructed here have free distance greater than the ones available in the literature.