Theory of additive complementary dual codes, constructions and computations

Official URL: https://dx.doi.org/10.1016/j.ffa.2023.102303

We study complementary dual Fq-linear codes defined over an extension field Fqm , which we refer to as additive complementary dual (ACD) codes. We first consider duality of this class of codes with respect to a general inner product on Fqm n, which covers the commonly studied cases such as the trace-Euclidean and Hermitian inner products. In general this inner product is not necessarily symmetric, hence we define the notions of left/right hulls and left/right duals of an additive code. This leads to one-sided ACD codes, for which we prove a characterization result in terms of the generator matrix, extending the analogous characterization of Massey for linear complementary dual (LCD) codes. Then we focus on constructions and parameters of ACD codes with respect to trace-Euclidean, Hermitian and Galois inner products. We prove how LCD codes yield ACD codes relative to inner products considered in this work, make some observations on MDS subclass of ACD codes and present a construction of an ACD code by expanding another ACD code. We provide extensive computational results and tables on ACD codes over F4,F8 and F9. Our computations yield many MDS ACD codes and also examples of ACD codes, which have more codewords than the comparable LCD codes with the same length and minimum distance.