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 >= [log(2)(3n + 1)] +epsilon n, where epsilon(n) = 1 if n = 8 4(m)-1/3 + {+/- 1, 2} or n = 4(m+2)-1/3 - {1, 2, 3} for some integer m >= 1 and epsilon(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 is not an element of {1, 2, 3} for even r and n <= 8(4(r-3) - 1)/3, 8(4(r-3) - 1)/3 - n not equal 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.
1-error correcting stabilizer codes optimal codes quantum error correction quantum hamming bound