Hamming distances of constacyclic codes of length 3ps and optimal codes with respect to the Griesmer and Singleton bounds

Abstract

Let p≠3 be a prime, s, m be positive integers, and λ be a nonzero element of the finite field Fpm. In [22] and [20], when the generator polynomials have one or two different irreducible factors, the Hamming distances of λ-constacyclic codes of length 3ps over Fpm have been considered. In this paper, we obtain that the Hamming distances of the repeated-root λ-constacyclic codes of length lps can be determined by the Hamming distances of the simple-root γ-constacyclic codes of length l, where l is a positive integer and λ=γps. Based on this result, the Hamming distances of the repeated-root λ-constacyclic codes of length 3ps are given when the generator polynomials have three different irreducible factors. Hence, the Hamming distances of all such constacyclic codes are determined. As an application, we obtain all optimal λ-constacyclic codes of length 3ps with respect to the Griesmer bound and the Singleton bound. Among others, several examples show that some of our codes have the best known parameters with respect to the code tables in [19].