Number of minimal cyclic codes with given length and dimension

TL;DR

This paper counts minimal cyclic codes of a given length and dimension over finite fields, linking the problem to counting irreducible factors of certain polynomials under specific prime factor conditions.

Contribution

It provides a new method to count minimal cyclic codes by analyzing irreducible factors of polynomials for particular prime factorizations of the code length.

Findings

Derived formulas for the number of minimal cyclic codes under specific prime factor conditions

Established the equivalence between counting minimal cyclic codes and irreducible polynomial factors

Extended understanding of the structure of cyclic codes in finite fields

Abstract

In this article, we count the quantity of minimal cyclic codes of length $n$ and dimension $k$ over a finite field $\mathbb F_q$, in the case when the prime factors of $n$ satisfy a special condition. This problem is equivalent to count the quantity of irreducible factors of $x^n-1\in \mathbb F_q[x]$ of degree $k$.