New Classes of Quantum Codes Associated with Surface Maps

If the cyclic sequences of {face types} {at} all vertices in a map are the same, then the map is said to be a semi-equivelar map. In particular, a semi-equivelar map is equivelar if the faces are the same type. Homological quantum codes represent a subclass of topological quantum codes. In this article, we introduce {thirteen} new classes of quantum codes. These codes are associated with the following: (i) equivelar maps of type $[k^k]$, (ii) equivelar maps on the double torus along with the covering of the maps, and (iii) semi-equivelar maps on the surface of \Echar{-1}, along with {their} covering maps. The encoding rate of the class of codes associated with the maps in (i) is such that $\frac{k}{n}\rightarrow 1$ as $n\rightarrow\infty$, and for the remaining classes of codes, the encoding rate is $\frac{k}{n}\rightarrow α$ as $n\rightarrow \infty$ with $α< 1$.

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