Repeated-root constacyclic codes of length 3lps and their dual codes

Abstract

Let p≠3 be any prime and l≠3 be any odd prime with gcd⁡(p,l)=1. The multiplicative group Fq⁎=〈ξ〉 can be decomposed into mutually disjoint union of gcd⁡(q−1,3lps) cosets over the subgroup 〈ξ3lps〉, where ξ is a primitive (q−1)th root of unity. We classify all repeated-root constacyclic codes of length 3lps over the finite field Fq into some equivalence classes by this decomposition, where q=pm, s and m are positive integers. According to these equivalence classes, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length 3lps over Fq and their dual codes. Self-dual cyclic codes of length 3lps over Fq exist only when p=2. We give all self-dual cyclic codes of length 3⋅2sl over F2m and their enumeration. We also determine the minimum Hamming distance of these codes when gcd⁡(3,q−1)=1 and l=1.