Topological hypermaps can be used to construct quantum stabilizer codes, although these codes are later shown to be just surface codes. In this article, we explore the quantum codes stem from the dual hypermaps and their relations with those from the original hypermaps. We find that the cell strucure of the surface codes from the dual hypermaps are just the dual cell structure of original hypermap itself with special darts erased. Then we propose a new kind of dual hypermaps called $\triangle$ -duals, and also the concept of their contrary maps. We also propose an alternative route to construct quantum codes from hypermaps. With all these new constructions, we find that the homological code of the original hypermap and that of the dual hypermap can be transformed to the new quantum codes of the $\triangle$ -dual and its contrary map, which makes the duality between these two codes more apparent both geometrically and algebraically.Topological hypermaps can be used to construct quantum stabilizer codes, although these codes are later shown to be just surface codes. In this article, we explore the quantum codes stem from the dual hypermaps and their relations with those from the original hypermaps. We find that the cell strucure of the surface codes from the dual hypermaps are just the dual cell structure of original hypermap itself with special darts erased. Then we propose a new kind of dual hypermaps called $\triangle$-duals, and also the concept of their contrary maps. We also propose an alternative route to construct quantum codes from hypermaps. With all these new constructions, we find that the homological code of the original hypermap and that of the dual hypermap can be transformed to the new quantum codes of the $\triangle$-dual and its contrary map, which makes the duality between these two codes more apparent both geometrically and algebraically.

May 4, 20210 citations

On the duals of quasi-cyclic codes and their application to quantum codes

Shivanshu Benjwal, Maheshanand Bhaintwal

Quantum Information ProcessingMar 23, 20243 citations

\triangle

-duals, and also the concept of their contrary maps. We also propose an alternative route to construct quantum codes from hypermaps. With all these new constructions, we find that the homological code of the original hypermap and that of the dual hypermap can be transformed to the new quantum codes of the

\triangle

-dual and its contrary map, which makes the duality between these two codes more apparent both geometrically and algebraically.