The object of this paper is to study two very important classes of codes in coding theory, namely self-orthogonal (SO) and linear complementary dual (LCD) codes under the symplectic inner product, involving characterization, constructions, and their application. Using such a characterization, we determine the mass formulas of symplectic SO and LCD codes by considering the action of the symplectic group, and further obtain some asymptotic results. Finally, under the Hamming distance, we obtain some symplectic SO (resp. LCD) codes with improved parameters directly compared with Euclidean SO (resp. LCD) codes. Under the symplectic distance, we obtain some additive SO (resp. additive complementary dual) codes with improved parameters directly compared with Hermitian SO (resp. LCD) codes. Further, we also construct many good additive codes outperform the best-known linear codes in Grassl's code table. As an application, we construct a number of record-breaking (entanglement-assisted) quantum error-correcting codes, which improve Grassl's code table.

翻译：本文研究编码理论中两类非常重要的码——辛内积下的自正交(SO)码与线性互补对偶(LCD)码，涉及其刻画、构造及应用。利用这种刻画，我们通过考虑辛群的作用确定了辛SO码与LCD码的质量公式，并进一步获得了一些渐近结果。最后，在汉明距离下，我们直接与欧几里得SO码（相应地，LCD码）比较，获得了一些参数改进的辛SO码（相应地，LCD码）。在辛距离下，我们直接与埃尔米特SO码（相应地，LCD码）比较，获得了一些参数改进的加法SO码（相应地，加法互补对偶码）。此外，我们还构造了许多优于Grassl码表中已知最优线性码的优良加法码。作为应用，我们构造了大量打破纪录的（纠缠辅助）量子纠错码，改进了Grassl码表。