There has been a lot of effort to construct good quantum codes from the classical error correcting codes. Constructing new quantum codes, using Hermitian self-orthogonal codes, seems to be a difficult problem in general. In this paper, Hermitian self-orthogonal codes are studied from algebraic function fields. Sufficient conditions for the Hermitian self-orthogonality of an algebraic geometry code are presented. New Hermitian self-orthogonal codes are constructed from projective lines, elliptic curves, hyper-elliptic curves, Hermitian curves, and Artin-Schreier curves. In addition, over the projective lines, we construct new families of MDS quantum codes with parameters $[[N,N-2K,K+1]]_q$ under the following conditions: i) $N=t(q-1)+1$ or $t(q-1)+2$ with $t|(q+1)$ and $K=\lfloor\frac{t(q-1)+1}{2t}\rfloor+1$; ii) $(n-1)|(q^2-1)$, $N=n$ or $N=n+1$, $K_0=\lfloor\frac{n+q-1}{q+1}\rfloor$, and $K\ge K_0+1$; iii) $N=tq+1$, $\forall~1\le t\le q$ and $K=\lfloor\frac{tq+q-1}{q+1}\rfloor+1$; iv) $n|(q^2-1)$, $n_2=\frac{n}{\gcd (n,q+1)}$, $\forall~ 1\le t\le \frac{q-1}{n_2}-1$, $N=(t+1)n+2$ and $K=\lfloor \frac{(t+1)n+1+q-1}{q+1}\rfloor+1$.
翻译:在古典错误校正代码中, 为构建良好的量子代码做出了大量的努力。 建立新的量子代码, 使用 Hermitian 自己或正反调代码, 似乎是一个一般的难题 。 在本文中, 正在从代数功能字段中研究 Hermitian 自己或正反调代码 。 展示了 Hermitian 自己或正正反调代码的足够条件 。 新的 Hermitian 自己或正反调代码是用投影行、 椭圆曲线、 超电子曲线、 Hermitian 曲线和 Artin- Schreier 曲线 。 此外, 在投影行外, 我们根据参数 $[ N, N-2K, K+1]_q美元; 美元= t( q) t (q), ⁇ 1+1美元; ⁇ * ⁇ +1美元; (q) ci=1 美元和 美元; k n==美元; k= 美元; n=x=美元; k=美元; k=美元; 美元; k=xxxxxxxx;