On the Existence of Hermitian Self-Dual Extended Abelian Group Codes

Abstract

Split group codes are a class of group algebra codes over an abelian group. They were introduced by Ding et al. (IEEE Trans. Inform. Theory IT-46:485–495, 2000) as a generalization of the cyclic duadic codes. For a prime power q and an abelian group G of order n such that gcd(n, q) = 1, consider the group algebra $\mathbb{F}_{q^{2}}[G^{{\ast}}]$ of $\mathbb{F}_{q^{2}}$ over the dual group G ^∗ of G. We prove that every ideal code in $\mathbb{F}_{q^{2}}[G^{{\ast}}]$ whose extended code is Hermitian self-dual is a split group code. We characterize the orders of finite abelian groups G for which an ideal code of $\mathbb{F}_{q^{2}}[G^{{\ast}}]$ whose extension is Hermitian self-dual exists and derive asymptotic estimates for the number of non-isomorphic abelian groups with this property.