New Qubit Codes from Multidimensional Circulant Graphs

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 ${\text{F}}_4$ codes from multidimensional circulant (MDC) graphs. We completely classify MDC graph codes for lengths $4\le n\le 40$ and show that many optimal $\text{dsb}\mathrm{\ell },0,d$ 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.