Results 11 - 20
of
50
A Class of Quantum LDPC Codes Derived from Latin Squares and Combinatorial Design
, 2007
"... Low Density Parity Check (LDPC) codes derived from Latin squares. The parity check matrices of these codes are constructed by permuting shift-orthogonal Latin squares of order n in blockrows and block-columns. I show that the constructed LDPC codes are self-orthogonal and their minimum and stopping ..."
Abstract
-
Cited by 5 (0 self)
- Add to MetaCart
(Show Context)
Low Density Parity Check (LDPC) codes derived from Latin squares. The parity check matrices of these codes are constructed by permuting shift-orthogonal Latin squares of order n in blockrows and block-columns. I show that the constructed LDPC codes are self-orthogonal and their minimum and stopping distances are bounded. This helps us to construct a family of quantum LDPC block codes. Consequently, we demonstrate that these constructed codes have good error correction capabilities and can be decoded using iterative decoding algorithms similar to their classical counterpart. I.
On subsystem codes beating the Hamming and Singleton bound
- Proc. Royal Soc. Series A
"... Subsystem codes are a generalization of noiseless subsystems, decoherence free subspaces, and quantum error-correcting codes. We prove a Singleton bound for Fqlinear subsystem codes. It follows that no subsystem code over a prime field can beat the Singleton bound. On the other hand, we show the rem ..."
Abstract
-
Cited by 4 (2 self)
- Add to MetaCart
(Show Context)
Subsystem codes are a generalization of noiseless subsystems, decoherence free subspaces, and quantum error-correcting codes. We prove a Singleton bound for Fqlinear subsystem codes. It follows that no subsystem code over a prime field can beat the Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the Hamming bound. A number of open problems concern the comparison in performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin’s work asks whether a subsystem code can use fewer syndrome measurements than an optimal MDS stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.
On the Classification of Hermitian Self-Dual Additive Codes over GF(9)
, 2011
"... Additive codes over GF(9) that are self-dual with respect to the Hermitian trace inner product have a natural application in quantum information theory, where they correspond to ternary quantum error-correcting codes. However, these codes have so far received far less interest from coding theorists ..."
Abstract
-
Cited by 3 (0 self)
- Add to MetaCart
Additive codes over GF(9) that are self-dual with respect to the Hermitian trace inner product have a natural application in quantum information theory, where they correspond to ternary quantum error-correcting codes. However, these codes have so far received far less interest from coding theorists than self-dual additive codes over GF(4), which correspond to binary quantum codes. Self-dual additive codes over GF(9) have been classified up to length 8, and in this paper we extend the complete classification to codes of length 9 and 10. The classification is obtained by using a new algorithm that combines two graph representations of self-dual additive codes. The search space is first reduced by the fact that every code can be mapped to a weighted graph, and a different graph is then introduced that transforms the problem of code equivalence into a problem of graph isomorphism. By an extension technique, we are able to classify all optimal codes of length 11 and 12. There are 56 005 876 (11, 3 11, 5) codes and 6493 (12, 3 12, 6) codes. We also find the smallest codes with trivial automorphism group.
CSS-like constructions of asymmetric quantum codes
- IEEE TRANS. INF. THEORY
, 2013
"... Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of q-ary AQCs is extended by removing the Fq-linearity requirement as well as the limit ..."
Abstract
-
Cited by 3 (1 self)
- Add to MetaCart
Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction of q-ary AQCs is extended by removing the Fq-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear pro-gramming, many optimal and good pure q-ary CSS-like codes for q ∈ {2, 3, 4, 5, 7, 8, 9} up to reasonable lengths are found. In many instances, removing the Fq-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.
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 co-subsystem while maintaining or improving the minimum distance. As a consequence, it is shown that all pure MDS subsystem codes are derived from MDS stabilize ..."
Abstract
-
Cited by 2 (2 self)
- Add to MetaCart
A generic method to derive subsystem codes from existing subsystem codes is given that allows one to trade the dimensions of subsystem and co-subsystem 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 error-correcting 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 error-correction. In this paper we give a brief exposi ..."
Abstract
-
Cited by 2 (2 self)
- Add to MetaCart
Recently, the field of quantum error-correcting 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 error-correction. 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.
A construciton of quantum stabilizer codes based on syndrome assignment by classical parity-check matrices
- IEEE Trans. Info. Theo
, 2011
"... ar ..."
(Show Context)
