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 ≥ ⌈log₂ (3n + 1)⌉ + ∈_n, where ∈_n = 1 if n = 84^m-1/3 + {±1, 2} or n = 4^m+2-1/3 -{1, 2, 3} for some integer m ≥ 1 and ∈_n = 0 otherwise. Or equivalently, a code [[n, n - r, 3]] exists if and only if n ≤ (4^r - 1)/3, (4^r - 1)/3 - n ∉ {1, 2, 3} for even r and n ≤ 8 (4^r-3 -1)/3, 8(4^-3 - 1)/3 - n ≠ 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.