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Essential Idempotents in Group Algebras and Minimal Cyclic Codes

Let G be a finite group of order n, Fq the field with q elements and assume that (n, q) = 1. Let e be an idempotent in Fq. For a normal subgroup H of G set Ĥ = 1/|H| ∑ h∈H h. If eĤ = e then FqG.e ⊂ FqGĤ and it is easy to see that, from the point of view of coding theory, this impies that the code defined by the ideal FqG.e is a repetition code. A primitive idempotent of FqG is called essential if this does not happen; i.e. if eĤ = 0 for every normal subgroup H of G. This idempotents were first considered by Bakshi, Raka and A. Sharma in [1], where they are called non-degenerate. If G is abelian, then G contains an essential idempotent if and only if G is cyclic. We shall give several algebraic characterizations of essential idempotents and compute their number in FqCn, where Cn denotes the cyclic group of order n. Using this concept, we are able to prove that every minimal abelian code is combinatorially equivalent to a minimal cyclic code. Finally, if we denote by m the order of q in Zn, set N = q − 1 and ` = N/n we establish a correspondence between essential idempotents of FqCn and those of FqCN . These results are joint work with R. Ferraz and G. Chalom [2]