Weight distributions of all irreducible $\mu$-constacyclic codes of length $\ell^n$

TL;DR

This paper determines the weight distributions and provides explicit generator polynomials for all irreducible -constacyclic codes of length ^n over finite fields, advancing understanding of their algebraic and combinatorial properties.

Contribution

It explicitly characterizes the weight distributions and generator polynomials of all such irreducible constacyclic codes, a previously unresolved problem.

Findings

Explicit weight distributions for all irreducible -constacyclic codes of length ^n

Closed-form expressions for generator polynomials and codewords

Enhanced understanding of code structure and properties

Abstract

Let $\mathbb{F}_q$ be a finite field of order $q$ and integer $n\ge 1$. Let $\ell$ be a prime such that $\ell^k|(q-1)$ for some integer $k\ge 1$ and $\mu$ be an element of order $\ell^k$ in $\mathbb{F}_q$. In this paper, we determine the weight distributions of all irreducible $\mu$-constacyclic codes of length $\ell^n$ over $\mathbb{F}_q$. Explicit expressions for the generator polynomials and codewords of these codes are also obtained.