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

TL;DR

This paper classifies and explicitly constructs all repeated-root constacyclic codes of length 3lp^s over finite fields, determines their duals, and characterizes conditions for self-duality, especially over fields with characteristic 2.

Contribution

It provides a complete classification, explicit generator polynomials, and dual code descriptions for these codes, including the enumeration of self-dual codes over fields with characteristic 2.

Findings

Classification of all such codes into equivalence classes.

Explicit generator polynomials for the codes and their duals.

Existence and enumeration of self-dual codes only when p=2.

Abstract

Let $p\neq3$ be any prime and $l\neq3$ be any odd prime with $gcd(p,l)=1$. $F_{q}^{*}=\langle\xi\rangle$ is decomposed into mutually disjoint union of $gcd(q-1,3lp^{s})$ coset over the subgroup $\langle\xi^{3lp^{s}}\rangle$, where $\xi$ is a primitive $(q-1)$th root of unity. We classify all repeated-root constacyclic codes of length $3lp^{s}$ 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 $3lp^{s}$ over $F_{q}$ and their dual codes. Self-dual cyclic(negacyclic) codes of length $3lp^{s}$ over $F_{q}$ exist only when $p=2$. And we give all self-dual cyclic(negacyclic) codes of length $3l2^{s}$over $F_{2^{m}}$ and its enumeration.