Several Families of Irreducible Constacyclic and Cyclic Codes

TL;DR

This paper introduces new families of irreducible constacyclic and cyclic codes over finite fields, detailing their weight distributions, duals, and optimal parameters, thereby expanding the understanding of their structure and properties.

Contribution

It constructs several new families of irreducible constacyclic and cyclic codes with specific weight distributions and optimal parameters, complementing existing research.

Findings

Identified families of irreducible constacyclic codes with few weights.

Constructed optimal constacyclic codes with specific parameters.

Presented new irreducible cyclic codes over finite fields.

Abstract

In this paper, several families of irreducible constacyclic codes over finite fields and their duals are studied. The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled. Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed. As by-products, a family of $[2n, (n-1)/2, d \geq 2(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ and a family of $[(q-1)n, (n-1)/2, d \geq (q-1)(\sqrt{n}+1)]$ irreducible cyclic codes over $\gf(q)$ are presented, where $n$ is a prime such that $\ord_n(q)=(n-1)/2$. The results in this paper complement earlier works on irreducible constacyclic and cyclic codes over finite fields.