All the stabilizer codes of distance 3

Abstract

We give necessary and sufficient conditions for the existence of stabilizer codes $[[n,k,3]]$ of distance 3 for qubits: $n-k\ge \lceil {\log }_2(3n+1)\rceil +{\epsilon }_n$ where ${\epsilon }_n=1$ if $n=8\frac{4^m-1}{3}+\{\pm 1,2\}$ or $n=\frac{4^{m+2}-1}{3}-\{1,2,3\}$ for some integer $m\ge 1$ and ${\epsilon }_n=0$ otherwise. Or equivalently, a code $[[n,n-r,3]]$ exists if and only if $n\le (4^r-1)/3,(4^r-1)/3-n\notin \{1,2,3\}$ for even $r$ and $n\le 8(4^{r-3}-1)/3,8(4^{r-3}-1)/3-n\ne 1$ 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.

7 pages with 8 tables