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We study additive quaternary codes whose parameters are close to those of the extended cyclic $[{12,6,6}]_{4}$ -code or to the quaternary linear codes generated by the elliptic quadric in $PG(3,4)$ or its dual. In particular we characterize those codes in the category of additive codes and construct some additive codes whose parameters are better than those of any linear quaternary code. Our new code parameters are $[{22,17.5,4}]_{4}$. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]