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.

翻译：众所周知，加法码可能具有比线性码更优的参数。然而，如何高效构造优于线性码的加法码，特别是构造比相同长度和维数的线性码具有更大距离的加法码，仍然是一个具有挑战性的问题。为解决该问题，本文聚焦于利用准循环码和组合方法构造优于线性码的加法码。首先，我们提出了偶数索引单生成元准循环码的最小辛距离的下界。进一步，通过计算机辅助组合与搜索方法，我们获得了大量具有大辛距离的二元准循环码，这些码均对应良好的四元加法码。值得注意的是，其中15种加法码在相同长度和维数下，其距离大于Grassl码表（四元线性码最小距离界，网址：http://www.codetables.de）中已知最优的四元线性码。此外，采用组合方法，我们部分确定了长度为$28$至$254$的最优四元加法3.5维码的参数。最后，作为扩展，我们还构造了许多良好的加法对偶码，其距离大于文献中已知最优的四元线性对偶码。