$G$-Equivalence in Group Algebras and Minimal Abelian Codes

Abstract

Let G be a finite abelian group and F a field such that char(F) | |G|. Denote by FG the group algebra of G over F. A (semisimple) abelian code is an ideal of FG. Two codes I1 and I2 of FG are G-equivalent if there exists an automorphism ψ of G whose linear extension to FG maps I1 onto I2.In this paper we give a necessary and sufficient condition for minimal abelian codes to be G-equivalent and show how to correct some results in the literature.Index Terms-group algebra, G-equivalence, primitive idempotent, abelian codes.

Group algebras and coding theory

“…For non-cyclic abelian groups, we may also apply the ideas above to construct idempotents. In [27], the following results are presented in details.…”

“…The study of the G-equivalence of ideals involves to know how the group of automorphisms Aut(G) acts on the lattice of the subgroups of G and hence on the idempotents in the group algebra which arise from these subgroups. From now on, we use the same notation for an automorphism of the group G and its linear extension to the group algebra F q G. The following results from [27] relate subgroups in G and idempotents in F q G. For finite abelian groups, Propositions 5.9, 5.10 and 5.11 below establish a correspondence between G-equivalent minimal ideals in F q G and Gisomorphic subgroups of G. Let G be a finite abelian group and F q a field such that char(F q ) | |G|. If e ′ , e ′′ ∈ P(F q G) are both different from G and H e ′ = H e ′′ , then there exists an automorphism ψ ∈ LAut(G) whose linear extension to F q G maps e ′ to e ′′ .…”

“…[27, Proposition V.3] Let m and r be positive integers and p a prime number. If G = (C p r ) m is a finite abelian p-group and F q is a field of char(F q ) = p. Then a primitive idempotent of…”

“…[27, Corollary V.4] Let m and r be positive integers, p a prime number, a finite abelian p-group G = (C p r ) m and F q a finite field with q elements such that o(q) = φ(p r ) in U(Z p r ). Then the minimal abelian codes in F q G are as follows, where h and K are as in Proposition 5.3.…”

Essential idempotents and simplex codes

“…Since in the case when char(F) | |A| the group algebra FA is semisimple and all ideals are direct sums of the minimal ones, it is only natural to study minimal abelian code -or -equivalently, primitive idempotents -and these has been done by several authors (see, for example [6], [5], [4] [9] [12] [14]). Also, Sabin and Lomonaco [15] have shown that central codes in metacyclic group algebras are equivalent to abelian codes.…”