Classes of quantum codes derived from self-dual orientable embeddings of complete multipartite graphs

Abstract

This paper presents four classes of binary quantum codes with minimum distance 3 and 4, namely Class-I, Class-II, Class-III and Class-IV. The classes Class-I and Class-II are constructed based on self-dual orientable embeddings of the complete graphs $K_{4r+1}$ and $K_{4s}$ and by current graphs and rotation schemes. The parameters of two classes of quantum codes are $[[2r(4r+1),2r(4r-3),3]]$ and $[[2s(4s-1),2(s-1)(4s-1),3]]$, respectively, where $r\ge 1$ and $s\ge 2$. For these quantum codes, the code rate approaches 1 as r and s tend to infinity. The Class-III with minimum distance 4 is constructed by using self-dual embeddings of complete bipartite graphs. The parameters of this class are $\left[ \left[ rs,\frac{(r-2)(s-2)}{2},4\right] \right]$, where r and s are both divisible by 4. The proposed Class-IV is the minimum distance 3 and code length $n=(2r+1)s^{2}$. This class is constructed based on self-dual embeddings of complete tripartite graph $K_{rs,s,s}$, and its parameters are $[[(2r+1)s^{2},(rs-2)(s-1),3]]$, where $r\ge 2$ and $s\ge 2$.