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 some 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，加性码可能具有比线性码更优的参数。然而，高效构造优于线性码的加性码（尤其是那些在相同长度和维数下具有更大距离的码）仍是一个具有挑战性的问题。为推进该问题，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性码。首先，我们提出了偶指标单生成元准循环码最小辛距离的下界。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，这些码均对应优良的四元加性码。值得注意的是，有15种加性码在Grassl码表（四元线性码最小距离界，网址：http://www.codetables.de）中，与相同长度和维数下已知最优的四元线性码相比，具有更大的距离。此外，采用组合方法，我们部分确定了长度为28至254的最优四元加性3.5维码的参数。最后，作为扩展，我们还构造了一些优良的加性互补对偶码，其距离大于文献中已知最优的四元线性互补对偶码。