A discussion of open problems concludes this paper. Bounds on the maximal length of maximum distance separable stabilizer codes are given. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Codes are given which map 3 qubits to 8 qubits correcting 1 error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors, and 1 to 29 qubits. Codes are given which map 3 qubits to 8 qubits correcting 1 error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors, and 1 to 29 qubits correcting 5 errors. Such states, useful for quantum teleportation and quantum error correction, are known for the. Abstract: A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. Abstract: A group theoretic framework is introduced that simplifies the description of known quantum error-correcting codes and greatly facilitates the construction of new examples. ![]() Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits. This paper also derives lower and upper bounds on the minimum distance of stabilizer codes, gives several code constructions, and derives numerous families of stabilizer codes, including quantum Hamming codes, quadratic residue codes, quantum Melas codes, quantum Bose-Chaudhuri-Hocquenghem (BCH) codes, and quantum character codes. Geometry of quantum states and quantum entanglement. over the eld GF (4) which are self-orthogonal with respect to a certain trace inner product. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over Fq 2 is provided that generalizes the well-known notion of additive codes over F4 of the binary case. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. This paper describes the basic theory of stabilizer codes over finite fields. Fault-tolerant quantum computation prompted the study of nonbinary quantum codes, but the theory of such codes is not as advanced as that of binary quantum codes. ![]() To address this difficulty, many good quantum error-correcting codes have been derived as binary stabilizer codes. Abstract: One formidable difficulty in quantum communication and computation is to protect information-carrying quantum states against undesired interactions with the environment.
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