Two new qubit stabilizer codes with parameters $[77, 0, 19]_2$ and $[90, 0, 22]_2$ are constructed for the first time by employing additive symplectic self-dual $\F_4$ codes from multidimensional circulant (MDC) graphs. We completely classify MDC graph codes for lengths $4\le n \le 40$ and show that many optimal $\dsb{\ell, 0, d}$ qubit codes can be obtained from the MDC construction. Moreover, we prove that adjacency matrices of MDC graphs have nested block circulant structure and determine isomorphism properties of MDC graphs.

翻译：本文首次通过使用来自多维循环（MDC）图的加法辛自对偶$\F_4$码，构造了两个新的量子比特稳定子码，参数分别为$[77, 0, 19]_2$和$[90, 0, 22]_2$。我们对长度$4\le n \le 40$的MDC图码进行了完全分类，并表明从MDC构造中可以获得许多最优的$\dsb{\ell, 0, d}$量子比特码。此外，我们证明了MDC图的邻接矩阵具有嵌套块循环结构，并确定了MDC图的同构性质。