Repeated-root constacyclic codes of length $3l{p}^s$ and their dual codes

Abstract

Let $p\ne 3$ be any prime and $l\ne 3$ be any odd prime with $gcd(p,l)=1$. $F_q^{\ast }=⟨\xi ⟩$ is decomposed into mutually disjoint union of $gcd(q-1,3l{p}^s)$ coset over the subgroup $⟨{\xi }^{3l{p}^s}⟩$, where $\xi$ is a primitive $(q-1)$th root of unity. We classify all repeated-root constacyclic codes of length $3l{p}^s$ over the finite field $F_q$ into some equivalence classes by the decomposition, where $q=p^m$, $s$ and $m$ are positive integers. According to the equivalence classes, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length $3l{p}^s$ over $F_q$ and their dual codes. Self-dual cyclic(negacyclic) codes of length $3l{p}^s$ over $F_q$ exist only when $p=2$. And we give all self-dual cyclic(negacyclic) codes of length $3l{2}^s$over $F_{2^m}$ and its enumeration.