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33
Nonbinary stabilizer codes over finite fields
 IEEE Trans. Inform. Theory
, 2006
"... One formidable difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. In past years, many good quantum errorcorrecting codes had been derived as binary stabilizer codes. Faulttolerant quantum comput ..."
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Cited by 50 (11 self)
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One formidable difficulty in quantum communication and computation is to protect informationcarrying quantum states against undesired interactions with the environment. In past years, many good quantum errorcorrecting codes had been derived as binary stabilizer codes. Faulttolerant 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. This paper describes the basic theory of stabilizer codes over finite fields. The relation between stabilizer codes and general quantum codes is clarified by introducing a Galois theory for these objects. A characterization of nonbinary stabilizer codes over Fq in terms of classical codes over F q 2 is provided that generalizes the wellknown notion of additive codes over F4 of the binary case. This paper 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 BCH codes, and quantum character codes. The puncturing theory by Rains is generalized to additive codes that are not necessarily pure. Bounds on the maximal length of maximum distance separable stabilizer codes are given. A discussion of open problems concludes this paper. 1
Nonbinary quantum ReedMuller codes
 In Proc. Int. Symp. Inform. Theory
, 2005
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Graphbased classification of selfdual additive codes over finite fields
 Adv. Math. Commun
, 2009
"... Abstract. Quantum stabilizer states over Fm can be represented as selfdual additive codes over F m 2. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to equivalence classes of codes. We have previously used thi ..."
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Cited by 8 (4 self)
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Abstract. Quantum stabilizer states over Fm can be represented as selfdual additive codes over F m 2. These codes can be represented as weighted graphs, and orbits of graphs under the generalized local complementation operation correspond to equivalence classes of codes. We have previously used this fact to classify selfdual additive codes over F4. In this paper we classify selfdual additive codes over F9, F16, and F25. Assuming that the classical MDS conjecture holds, we are able to classify all selfdual additive MDS codes over F9 by using an extension technique. We prove that the minimum distance of a selfdual additive code is related to the minimum vertex degree in the associated graph orbit. Circulant graph codes are introduced, and a computer search reveals that this set contains many strong codes. We show that some of these codes have highly regular graph representations. 1.
Quantum block and convolutional codes from selforthogonal product codes
 in Proc. 2005 IEEE Intl. Symp. Inform. Theory
, 2005
"... Abstract — We present a construction of selforthogonal codes using product codes. From the resulting codes, one can construct both block quantum errorcorrecting codes and quantum convolutional codes. We show that from the examples of convolutional codes found, we can derive ordinary quantum error ..."
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Abstract — We present a construction of selforthogonal codes using product codes. From the resulting codes, one can construct both block quantum errorcorrecting codes and quantum convolutional codes. We show that from the examples of convolutional codes found, we can derive ordinary quantum errorcorrecting codes using tailbiting with parameters [42N, 24N, 3]2. While it is known that the product construction cannot improve the rate in the classical case, we show that this can happen for quantum codes: we show that a code [15, 7,3]2 is obtained by the product of a code [5, 1, 3]2 with a suitable code. I.
Quantum convolutional codes derived from reedsolomon and reedmuller codes,” arXiv:quantph/0701037
, 2007
"... Abstract — Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. Two families of quantum convolutional cod ..."
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Cited by 5 (1 self)
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Abstract — Convolutional stabilizer codes promise to make quantum communication more reliable with attractive online encoding and decoding algorithms. This paper introduces a new approach to convolutional stabilizer codes based on direct limit constructions. Two families of quantum convolutional codes are derived from generalized ReedSolomon codes and from ReedMuller codes. A Singleton bound for pure convolutional stabilizer codes is given. I.
The Weights in MDS Codes
, 908
"... Abstract—The weights in MDS codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n − k + 1 to n. The proof uses the covering radius o ..."
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Cited by 4 (2 self)
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Abstract—The weights in MDS codes of length n and dimension k over the finite field GF(q) are studied. Up to some explicit exceptional cases, the MDS codes with parameters given by the MDS conjecture are shown to contain all k weights in the range n − k + 1 to n. The proof uses the covering radius of the dual code. Index Terms—MDS codes, quantum codes, weight distribution, covering radius I.
Subsystem Code Constructions
, 2007
"... A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and cosubsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilize ..."
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Cited by 2 (2 self)
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A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and cosubsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilizer codes. The existence of numerous MDS subsystem codes is established. Another propagation rule is derived that allow one to obtain longer subsystem codes from a given subsystem code.
Nonbinary Stabilizer Codes
"... Recently, the field of quantum errorcorrecting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum errorcorrection. In this paper we give a brief exposi ..."
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Recently, the field of quantum errorcorrecting codes has rapidly emerged as an important discipline. As quantum information is extremely sensitive to noise, it seems unlikely that any large scale quantum computation is feasible without quantum errorcorrection. In this paper we give a brief exposition of the theory of quantum stabilizer codes. We review the stabilizer formalism of quantum codes, establish the connection between classical codes and stabilizer codes and the main methods for constructing quantum codes from classical codes. In addition to the expository part, we include new results that cannot be found elsewhere. Specifically, after reviewing some important bounds for quantum codes, we prove the nonexistence of pure perfect quantum stabilizer codes with minimum distance greater than 3. Finally, we illustrate the general methods of constructing quantum codes from classical codes by explicitly constructing two new families of quantum codes and conclude by showing how to construct new quantum codes by shortening.