All the stabilizer codes of distance 3

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$n=8\frac{4^m-1}3+{\pm1,2}$or$n=\frac{4^{m+2}-1}3-{1,2,3}$ for some integer $m\ge 1$ and ${ϵ}_n=0$ otherwise. Or equivalently, a code $[[n,n-r,3]]$ exists if and only if $n\leq (4^r-1)/3, (4^r-1)/3-n\notin\lbrace 1,2,3\rbrace$foreven$r$and$n\leq 8(4^{r-3}-1)/3, 8(4^{r-3}-1)/3-n\not=1$forodd$r$\mathrm{.}Givenanarbitrarylength$n$ we present an explicit construction for an optimal quantum stabilizer code of distance 3 that saturates the above bound.