Hermitian Self-Dual Abelian Codes

Somphong Jitman,San Ling,Patrick Sole

doi:10.1109/tit.2013.2296495

Abstract

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 = 2 l 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 ≅ A⊕Z 2 k with 2 ≠ |A|. Based on the characterization obtained, the number of Hermitian self-dual abelian codes in F 2 2l [A⊕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⊕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 ⊕ 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.

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