Weight enumerator of some irreducible cyclic codes

TL;DR

This paper explicitly determines the weight enumerators for all irreducible cyclic codes of certain lengths over finite fields, under specific divisibility conditions, providing comprehensive formulas for these codes.

Contribution

It provides explicit formulas for the weight enumerators of irreducible cyclic codes when prime divisors of length divide q-1, covering all such cases.

Findings

Explicit weight enumerators for all relevant irreducible cyclic codes.

Complete classification under the divisibility condition.

Facilitates analysis of code properties and error correction capabilities.

Abstract

In this article, we show explicitly all possible weight enumerators for every irreducible cyclic code of length $n$ over a finite field $\mathbb F_q$, in the case which each prime divisor of $n$ is also a divisor of $q-1$.