On the Hamming distances of repeated-root cyclic codes of length $5p^s$

TL;DR

This paper analyzes the weight distributions and Hamming distances of cyclic codes of length 5 and their repeated-root extensions of length 5p^s, identifying MDS codes and constructing quantum synchronizable codes.

Contribution

It determines the weight distributions and Hamming distances for a broad class of cyclic and repeated-root cyclic codes of length 5p^s, including MDS codes and quantum codes.

Findings

Complete weight distributions for cyclic codes of length 5.

Hamming distances for repeated-root cyclic codes of length 5p^s.

Identification of all MDS cyclic codes of length 5p^s.

Abstract

Due to the wide applications in consumer electronics, data storage systems and communication systems, cyclic codes have been an interesting research topic in coding theory. In this paper, let $p$ be a prime with $p\ge 7$. We determine the weight distributions of all cyclic codes of length $5$ over $\f_q$ and the Hamming distances of all repeated-root cyclic codes of length $5p^s$ over $\F_q$, where $q=p^m$ and both $s$ and $m$ are positive integers. Furthermore, we find all MDS cyclic codes of length $5p^s$ and take quantum synchronizable codes from repeated-root cyclic codes of length $5p^s$.