On Additive Quasi-abelian Codes over Finite Fields and Their Duality Properties

In this paper, we introduce and study a new class of additive codes over finite fields, viz. additive quasi-abelian (QA) codes, which is an extension of (linear) quasi-abelian codes over finite fields. We study algebraic structures of these codes and their dual codes with respect to ordinary and Hermitian trace bilinear forms. We further express these codes as direct sums of additive codes over finite fields and derive necessary and sufficient conditions under which an additive QA code is (i) self-orthogonal, (ii) self-dual, and (iii) an additive code with complementary dual (or an ACD code). We also derive necessary and sufficient conditions for the existence of a self-dual additive QA code over a finite field. Besides this, we obtain explicit enumeration formulas for all self-orthogonal, self-dual, and ACD additive QA codes over finite fields. We also list several MDS and almost MDS codes belonging to the family of additive QA codes.