New Qubit Codes from Multidimensional Circulant Graphs

TL;DR

This paper introduces new quantum stabilizer codes derived from multidimensional circulant graphs, providing a classification of such graphs and demonstrating their potential to produce optimal qubit codes with specific parameters.

Contribution

It presents the first construction of qubit codes from multidimensional circulant graphs and classifies these graphs for lengths up to 40, revealing their isomorphism properties.

Findings

Constructed new qubit codes with parameters [77, 0, 19]_2 and [90, 0, 22]_2.

Classified MDC graph codes for lengths 4 to 40.

Proved adjacency matrices of MDC graphs have nested block circulant structure.

Abstract

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.