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

作者：Dinh, Hai Q.; Wang, Xiaoqiang^*; Liu, Hongwei; Yamaka, Woraphon

来源：Finite Fields and their Applications, 2021, 70: 101794.

DOI：10.1016/j.ffa.2020.101794

• Constacyclic code
• Repeated-root code
• Hamming distance
• Griesmer bound
• Singleton bound

Let p not equal 3 be a prime, s, in be positive integers, and A be a nonzero element of the finite field F-pm . In [22] and [20], when the generator polynomials have one or two different irreducible factors, the Hamming distances of lambda-constacyclic codes of length 3ps over F-pm have been considered. In this paper, we obtain that the Hamming distances of the repeated root lambda-constacyclic codes of length lps can be determined by the Hamming distances of the simple-root gamma-constacyclic codes of length l, where l is a positive integer and lambda = gamma(ps). Based on this result, the Hamming distances of the repeated root lambda-constacyclic codes of length 3p(s) 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 lambda-constacyclic codes of length 3p(s) 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].