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 [log 2(3n+1)] +\epsilon n, where n=1 if n=8 4m-1 3+± 1,2\ or n= 4m+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 (4r-1)/3 (4r-1)/3-n\notin \lbrace 1,2,3\rbrace for even r and n 8(4 r-3-1)/3, 8(4r-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. © 1963-2012 IEEE.