Properties and Comparisons of Various Graphs and Their Codes

It is established in literature that finding stabilizer quantum error correcting codes (QECCS) is the same as finding self-dual additive codes over the finite field $\mathbb{F}_{4}$ under the Hermitian trace inner product. Additionally, every self-dual additive code can be represented by a graph adjacency matrix. Many selfdual additive codes are constructed from circulant graphs. We introduce new graph code constructions: the Toeplitz Graph, the Multidimensional Toeplitz Graph (MDT), and the Generalized Toeplitz (GT) graph constructions. We consider some of the properties of the Toeplitz and Multidimensional Toeplitz Graphs and compare the Circulant and GT code constructions.