AI 中文总结
研究自正交码，通过建立准则，结合部分差集、子群作用及特征函数法等构造新的自正交码族与极小码，得到最优或几乎最优量子码以及具有高参数灵活性的最优量子码，还构造了违反特定条件的自正交极小码。
AI 中文摘要
自正交码因其在量子纠错码、线性互补对偶码等众多领域的应用而备受关注。本文通过建立表征某些线性码自正交性的准则，构造了新的自正交码族和自正交极小码。首先，针对由定义集构造产生的线性码，建立了几个自正交性的新准则，结合部分差集构造了新的自正交码族并得到最优或几乎最优量子码。其次，通过乘法子群作用于射影线性码列构造自正交码并导出量子码，获得具有高参数灵活性的最优量子码。最后，利用特征函数法构造线性码并建立自正交性准则，构造了几类违反阿希克明 - 巴尔格条件的自正交极小码。
英文摘要
Self-orthogonal codes have attracted considerable attention owing to their applications in quantum error-correcting codes, linear complementary dual codes, and a variety of other fields. In this paper, we construct new families of self-orthogonal codes and self-orthogonal minimal codes by establishing criteria that characterize the self-orthogonality of certain linear codes. To this end, we first establish several new criteria for linear codes arising from the defining-set construction to be self-orthogonal. More specifically, for $q > 3$, we show that the code $\mathcal{C}_D$ is self-orthogonal whenever the defining set $D$ is $G$-invariant, where $G \subseteq \mathbb{F}_q^*$ and $|G| > 2$. For $q = 2, 3$, we characterize the self-orthogonality of $\mathcal{C}_D$ using certain character sums. By combining these criteria with partial difference sets, we construct several new families of self-orthogonal codes, which lead to optimal or almost optimal quantum codes. Secondly, via the action of a multiplicative subgroup $G \subseteq \mathbb{F}_q^*$ with $|G| > 2$ on the columns of a projective linear code, we construct self-orthogonal codes with flexible parameters. From their augmented codes, we derive quantum codes. By choosing suitable projective codes, we obtain optimal quantum codes with high parametric flexibility. Thirdly, we employ the characteristic function method to construct linear codes and establish criterion for their self-orthogonality. Based on this, we construct several classes of self-orthogonal minimal codes that violate the Ashikhmin-Barg condition, using vectorial dual-bent functions and $s$-plateaued functions.
Comments35 pages
