All the Stabilizer Codes of Distance 3
Sixia Yu, Jürgen Bierbrauer, Ying Dong, Qing Chen, C. H. Oh · IEEE Transactions on Information Theory · 2013
We give necessary and sufficient conditions for the existence of stabilizer codes$[[n,k,3]]$of distance 3 for qubits:$n-k\geq \lceil \log _{2}(3n+1)\rceil +\epsilon _{n}$, where$\epsilon _{n}=1$if$n=8 {{ 4^{m}-1}\over { 3}}+\{\pm 1,2\}$or$n= {{ 4^{m+2}-1}\over { 3}}-\{1,2,3\}$for some integer$m\geq 1$and$\epsilon _{n}=0$otherwise. Or equivalently, a code$[[n,n-r,3]]$exists if and only if$n\leq (4^{r}-1)/3, (4^{r}-1)/3-n otin \lbrace 1,2,3\rbrace $for even$r$and$n\leq 8(4^{r-3}-1)/3, 8(4^{r-3}-1)/3-n e 1$for odd$r$. Given an arbitrary length$n$, we present an explicit construction for an optimal quantum stabilizer code of distance 3 that saturates the above bound.