ACD codes over skew-symmetric dualities

Abstract

The applications of additive codes mainly lie in quantum error correction and quantum computing. Due to their applications in quantum codes, additive codes have grown in importance. In addition to this, additive codes allow the implementation of a variety of dualities. The article begins by developing the properties of Additive Complementary Dual (ACD) codes with respect to arbitrary dualities over finite abelian groups. Further, we introduce a subclass of non-symmetric dualities referred to as skew-symmetric dualities. Then, we precisely count symmetric and skew-symmetric dualities over finite fields. Two conditions have been obtained: one is a necessary and sufficient condition, and the other is a necessary condition. The necessary and sufficient condition is for an additive code to be an ACD code over arbitrary dualities. The necessary condition is on the generator matrix of an ACD code over skew-symmetric dualities. We provide bounds for the highest possible minimum distance of ACD codes over skew-symmetric dualities. Finally, we find some new quaternary ACD codes over non-symmetric dualities with better parameters than the symmetric ones.