Number of minimal cyclic codes with given length and dimension

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 $k$.