最优二元自正交码

Optimal Binary Self-orthogonal Codes

构造一般二元自正交码是经典纠错码和量子纠错码研究的难点。研究基于并置二元循环矩阵的1-生成子拟循环码结构。以向量移位等价、线性码等价以及二元自正交码码字偶重量特点等为基础,设计特殊二元拟循环码结构,构造了28个最优或已知最优二元拟循环自正交码。提出自正交码截短-删除方法,构造出所获得自正交码的62个衍生码。文中的90个二元自正交码与文献[13]中最优或已知最优线性码比较,分别有67和23个二元自正交码是最优和已知最优。构造结果验证2个方法对一般二元自正交码构造的有效性,同时能较好解决量子纠错码构造中具有尽可能大对偶重量自正交码的设计问题。

Designing general binary self-orthogonal codes is a difficult problem in both classical coding theory and quantum coding theory. The structure of one-generator quasi-cyclic codes constructed by concatenating binary circulant matrices is investigated. Twenty-eight optimal or best known binary self-orthogonal codes are built by designing the structure of a special subclass of quasi-cyclic codes, which takes advantage of some restrictions such as the shifting equivalence relation on vector, the equivalence relation on linear codes and even weight property of binary self-orthogonal codes. A puncturing-expurgating construction method for binary self-orthogonal codes is proposed, and sixty-two derived codes from these obtained self -orthogonal codes are constructed. In comparison with Literature （13）, 67 and 23 among our ninety self- orthogonal codes are separately optimal and best known. The construction results indicate that these two methods are effective to design general self-orthogonal codes. Furthermore, the ideas can preferably solve the construction problem of self-orthogonal codes with possible larger minimum dual weight, which is the critical infrastructure in designing better quantum codes.