Some good additive codes outperform linear counterparts

It is well known that additive codes may have better parameters than linear codes. However, it is still a challenging problem to efficiently construct additive codes that outperform linear codes, especially those with greater distance than linear codes of the same length and dimension. To advance this problem, this paper focuses on constructing additive codes that outperform linear codes using quasi-cyclic codes and combinatorial methods. Firstly, we propose a lower bound on the minimum symplectic distance of 1-generator quasi-cyclic codes of index even. Further, we get many binary quasi-cyclic codes with large symplectic distances utilizing computer-supported combination and search methods, all corresponding to good quaternary additive codes. Notably, 15 additive codes have greater distances than best-known quaternary linear codes in Grassl's code table (bounds on the minimum distance of quaternary linear codes http://www.codetables.de) for the same lengths and dimensions. Moreover, employing a combinatorial approach, we partially determine the parameters of optimal quaternary additive 3.5-dimensional codes with lengths from $28$ to $254$. Finally, as an extension, we also construct many good additive complementary dual codes with larger distances than best-known quaternary linear complementary dual codes in the literature.