Hamming distances of constacyclic codes of length 7ps over Fpm

Abstract

In this paper, we study constacyclic codes of length $n=7p^s$ over a finite field of characteristics p, where $p\ne 7$ is an odd prime number and s a positive integer. The previous methods in the literature that were used to compute the Hamming distances of repeated-root constacyclic codes of lengths $n{p}^s$ with $1\le n\le 6$ cannot be applied to completely determine the Hamming distances of those with $n=7$. This is due to the high computational complexity involved and the large number of unexpected intermediate results that arise during the computation. To overcome this challenge, we propose a computer-assisted method for determining the Hamming distances of simple-root constacyclic codes of length 7, and then utilize it to derive the Hamming distances of the repeated-root constacyclic codes of length $7p^s$. Our method is not only straightforward to implement but also efficient, making it applicable to these codes with larger values of n as well. In addition, all self-orthogonal, dual-containing, self-dual, MDS and AMDS codes among them will also be characterized.