Idempotents Generators for Minimal Cyclic Codes of Length p ⁿ q

Abstract

Let p and q be distinct positive prime numbers and ℓ a positive integer such that $\mathop{\mathrm{gcd}}\nolimits (\ell,pq) = 1$. For a natural number n ≥ 1, let $\mathcal{C}_{p^{n}q}$ be a cyclic group of order p ⁿ q and $\mathbb{F}_{\ell}$ a finite field with ℓ elements. In this paper we explicitly present the primitive idempotents of the group algebra $\mathbb{F}_{\ell}\mathcal{C}_{p^{n}q}$ under some further restrictions on ℓ, p and q. These idempotents generate the minimal ideals of $\mathbb{F}_{\ell}\mathcal{C}_{p^{n}q}$, hence the minimal cyclic codes of length p ⁿ q. Our computation is based on techniques developed by Bakshi and Raka (Finite Fields Appl 9(4):432–448, 2003) and Ferraz and Polcino Milies (Finite Fields Appl 13:382–393, 2007). A particular example for codes of length 245 is computed and we believe that this points out some mistakes in current literature on this subject.

Work partially supported by CNPq/(Brasil).

Work partially supported by FAPEMIG/MG(Brasil).