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:
$n-k\geq \lceil \log _{2}(3n+1)\rceil +\epsilon _{n}$
, 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\notin \lbrace 1,2,3\rbrace $
for even
$r$
and
$n\leq 8(4^{r-3}-1)/3, 8(4^{r-3}-1)/3-n\ne 1$
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.