Hermitian Self-Dual Abelian Codes

Hermitian self-dual abelian codes in a group ring F (q)2[G], where F(q)2 is a finite field of order q(2) 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(q)2 [G] is given, together with an alternative proof of necessary and sufficient conditions for the existence of such a code in F(q)2 [G], i. e., there exists a Hermitian self-dual abelian code in F(q)2 [G] if and only if the order of G is even and q = 2l for some positive integer l. Later on, the study is further restricted to the case where F(2)2l [G] is a principal ideal group ring, or equivalently, G congruent to A circle plus Z(2)k with 2 vertical bar |A|. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in F(2)2l [A circle plus Z(2)k] 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, F(2)2l [A circle plus Z(2)k] 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 F(2)2l [A circle plus Z(2)] is established, together with the distribution of odd integers m such that a unique Hermitian self-dual cyclic code of length 2 m over F(2)2l exists.