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Erschienen in:
2015 | OriginalPaper | Buchkapitel
Idempotents Generators for Minimal Cyclic Codes of Length p n q
verfasst von : Gustavo Terra Bastos, Marinês Guerreiro
Erschienen in: Coding Theory and Applications
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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.