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 $$s\ge 2$$ .