$\mathbb{F}_{2}\mathbb{F}_{4}$-Additive Complementary Dual Codes

Abstract: In this paper, we investigate the structure and properties of additive complementary dual (ACD) codes over the mixed alphabet $\mathbb{F}2\mathbb{F}_4$ relative to a certain inner product defined over $\mathbb{F}_2\mathbb{F}_4$. We establish sufficient conditions under which such codes are additive complementary dual (ACD) codes. We also show that ACD codes over $\mathbb{F}{2}\mathbb{F}_{4}$ can be applied to construct binary linear complementary dual codes as their images under the linear map $W$. Notably, we prove that if the binary image of a code is LCD, then the original code is necessarily ACD. An example is given where the image is a distance-optimal binary LCD code.