Hermitian Self-Dual Abelian Codes

Hermitian self-dual abelian codes in a group ring F_q²[G], where F_q² is a finite field of order q² and G is a finite abelian group, are studied. Using the well-known discrete Fourier transform decomposition for a semisimple group ring, a characterization of Hermitian self-dual abelian codes in F_q²[G] is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in F_q²[G], i.e., there exists a Hermitian self-dual abelian code in F_q²[G] if and only if the order of G is even and q = 2^l for some positive integer l. Later on, the study is further restricted to the case where F₂^2l [G] is a principal ideal group ring, or equivalently, G ≅ A⊕Z₂k with 2 ≠ |A|. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in F₂2l [A⊕Z₂k] can be determined easily. When A is cyclic, this result answers an open problem of Jia et al. concerning Hermitian self-dual cyclic codes. In many cases, F₂2l [A⊕Z₂k] contains a unique Hermitian self-dual abelian code. The criteria for such cases are determined in terms of l and the order of A. Finally, the distribution of finite abelian groups A such that a unique Hermitian self-dual abelian code exists in F₂2l [A ⊕ Z₂] is established, together with the distribution of odd integers m such that a unique Hermitian self-dual cyclic code of length 2 m over F₂2l exists.