The unique case of Euclidean self-dual abelian codes in principal ideal group algebras
Somphong Jitman · 2016
As a generalization of self-dual cyclic codes, self-dual abelian codes form an important class of linear codes containing many good and practical codes. It is known that there exists a Euclidean self-dual abelian code in a principal ideal group algebra Fq[G] if and only if q = 2ℓand G = A × ℤ2k, where ℓ and k are positive integers and A is an abelian group of odd order. Based on the enumeration of Euclidean self-dual abelian codes in F2ℓ[A ⊕ ℤ2k], it turns out that, in many cases, F2ℓ[A ⊕ ℤ2k] contains a unique Euclidean self-dual abelian code. In this paper, we focus on this uniqueness property. Some sufficient conditions for F2ℓ[A ⊕ ℤ2k] to contain a unique Euclidean self-dual abelian code are given. Subsequently, the distribution of finite abelian groups A of odd order such that a unique Euclidean self-dual abelian code exists in F2ℓ[A ⊕ ℤ2K] is established and it turns out that the unique case occurs less frequently as |A| grows.