The 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 distances than linear codes of the same lengths and dimensions. This paper focuses on constructing additive codes that outperform linear codes based on quasi-cyclic codes and combinatorial methods. Firstly, we propose a lower bound on the symplectic distance of 1-generator quasi-cyclic codes of index even. Secondly, we get many binary quasi-cyclic codes with large symplectic distances utilizing computer-supported combination and search methods, all of which correspond to good quaternary additive codes. Notably, some 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 the best-known quaternary linear complementary dual codes in the literature.

翻译：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（尤其是具有比相同长度和维数线性码更大距离的加性码）仍是具有挑战性的问题。本文聚焦于基于准循环码和组合方法构造优于线性码的加性码。首先，我们提出了偶数索引单生成元准循环码的辛距离下界。其次，利用计算机支持的组合与搜索方法，我们获得了大量具有大辛距离的二元准循环码，所有这些码均对应良好的四元加性码。值得注意的是，部分加性码在Grassl码表（四元线性码最小距离界，http://www.codetables.de）中相同长度和维数下，具有比已知最佳四元线性码更大的距离。此外，采用组合方法，我们部分确定了长度从$28$到$254$的最优四元加性3.5维码的参数。最后，作为扩展，我们还构造了一些具有比文献中已知最佳四元线性对偶码更大距离的良好加性对偶码。