Title :
Quantum codes of minimum distance two
Author_Institution :
AT&T Res., Florham Park, NJ, USA
fDate :
1/1/1999 12:00:00 AM
Abstract :
It is reasonable to expect the theory of quantum codes to be simplified in the case of codes of minimum distance 2; thus it makes sense to examine such codes in the hopes that techniques that prove effective there will generalize. With this in mind, we present a number of results on codes of minimum distance 2. We first compute the linear programming bound on the dimension of such a code, then show that this bound can only be attained when the code either is of even length, or is of length 3 or 5. We next consider questions of uniqueness, showing that the optimal code of length 2 or 1 is unique (implying that the well-known one-qubit-in-five single-error correcting code is unique), and presenting nonadditive optimal codes of all greater even lengths. Finally, we compute the full automorphism group of the more important distance 2 codes, allowing us to determine the full automorphism group of any GF(4)-linear code
Keywords :
Galois fields; error correction codes; group codes; linear codes; linear programming; GF(4)-linear code; code dimension; even length code; full automorphism group; linear programming bound; minimum distance two; nonadditive optimal codes; one-qubit-in-five single-error correcting code; optimal code; quantum codes; uniqueness; Binary codes; Cryptography; Encoding; Linear programming; Quantum mechanics; Rain; Vectors;
Journal_Title :
Information Theory, IEEE Transactions on