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Problems of Information Transmission

On error correction with errors in both the channel and syndrome

Serge G. Vladut, Grigory A. Kabatiansky, V. V. Lomakov

Published Apr 1, 2015• 11 citations
Publisher

Abstract

We address the problem of error correction by linear block codes under the assumption that the syndrome of a received vector is found with errors. We propose a construction of parity-check matrices which allow to solve the syndrome equation even with an erroneous syndrome, in particular, parity-check matrices with minimum redundancy, which are analogs of Reed-Solomon codes for this problem. We also establish analogs of classical coding theory bounds, namely the Hamming, Singleton, and Gilbert-Varshamov bounds. We show that the new problem can be considered as a generalization of the well-known Ulam’s problem on searching with a lie and as a discrete analog of the compressed sensing problem.

DOI
10.1134/s0032946015020040

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We address the problem of error correction by linear block codes under the assumption that the syndrome of a received vector is found with errors. We propose a construction of parity-check matrices which allow to solve the syndrome equation even with an erroneous syndrome, in particular, parity-check matrices with minimum redundancy, which are analogs of Reed-Solomon codes for this problem. We also establish analogs of classical coding theory bounds, namely the Hamming, Singleton, and Gilbert-Varshamov bounds. We show that the new problem can be considered as a generalization of the well-known Ulam’s problem on searching with a lie and as a discrete analog of the compressed sensing problem.
Most decoding algorithms of linear codes, in general, are designed to correct or detect errors. However, many channels cause erasures in addition to errors. In principle, decoding over such channels can be accomplished by deleting the erased symbols and decoding the resulting vector with respect to a punctured code. For any given linear code and any given maximum number of correctable erasures, we introduce parity-check matrices yielding parity-check equations that do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures. This allows for the separation of erasures from errors to facilitate decoding. The parity-check matrices typically have redundant rows. We give several constructions of such matrices and prove general bounds on their minimum sizes.
Linear block codes over noisy channels causing both erasures and errors can be decoded by deleting the erased symbols and decoding the resulting vector with respect to a punctured code and then retrieving the erased symbols. This can be accomplished using separating parity-check matrices. For a given maximum number of correctable erasures, such matrices yield parity-check equations that do not check any of the erased symbols and which are sufficient to characterize all punctured codes corresponding to this maximum number of erasures. Separating parity-check matrices typically have redundant rows. An upper bound on the minimum number of rows in separating parity-check matrices, which is called the separating redundancy, is derived which proves that the separating redundancy tends to behave linearly as a function of the code length.
Abstract Since coding has become a basic tool for practically all communication/electronic devices, it is important to carefully study the error patterns that actually occur. This allows correction of only partial errors rather than those which have been studied using Hamming distance, in non-binary cases. The paper considers a class of distances, SK-distances, in terms of which partial errors can be defined. Examining the sufficient condition for the existence of a parity check matrix for a given number of parity-checks, the paper contains an upper bound on the number of parity check digits for linear codes with property that corrects all partial random errors of an (n, k ) code with minimum SK-distance at least d. The result generalizes the rather widely used Varshamov-Gilbert bound, which follows from it as a particular case.
Performing active quantum error correction to protect fragile quantum states highly depends on the correctness of measured error syndromes. To obtain reliable error syndromes using imperfect physical circuits, we propose syndrome measurement (SM) and quantum data-syndrome (DS) codes. SM codes protect syndrome with linearly dependent redundant stabilizer measurements. DS codes generalize this idea for simultaneous correction of both data qubits and syndrome bits errors. We study fundamental properties of quantum DS codes, including split weight enumerators, generalized MacWilliams identities, and linear programming bounds. In particular, we derive Singleton and Hamming-type upper bounds on the minimum distance of degenerate quantum DS codes. Then we study random DS codes and show that random DS codes with a relatively small additional syndrome measurements achieve the Gilbert-Varshamov bound of stabilizer codes. Finally, we propose a family of CSS-type quantum DS codes based on classical cyclic codes, which include the Steane code and the quantum Golay code.
In this paper, a new but simple construction of stabilizer codes and related entanglement-assisted quantum error-correcting codes is proposed based on syndrome assignment by classical parity-check matrices. This method turns the construction of quantum stabilizer codes to the construction of classical parity-check matrices satisfying a specific commutative condition. The designed minimum distance 2t*+1 of the constructed quantum stabilizer codes can be achieved by a commutative classical parity-check matrix with classical minimum distance 4t*-m, where the parameterm, 0 ≤m≤ 2t*, depends on a property of the parity-check matrix. Asmdecreases, there is an increasing set of additional correctable error operators beyond the designed error correcting capability t*. The (asymptotic) coding efficiency is at least comparable to that of CSS codes. A class of quantum Reed-Muller codes is constructed and codes in this class have a larger set of correctable error operators than that of the quantum Reed-Muller codes previously developed in the literature. Quantum circulant codes are also constructed and many of them are optimal in terms of their coding parameters.
We discuss single-shot decoding of quantum Calderbank-Shor-Steane codes with faulty syndrome measurements. We state the problem as a joint source-channel coding problem. By adding redundant rows to the code’s parity-check matrix, we obtain an additional syndrome error correcting code which addresses faulty syndrome measurements. Thereby, the redundant rows are chosen to obtain good syndrome error correction capabilities while keeping the stabilizer weights low. Optimal joint decoding rules are derived which, though too complex for general codes, can be evaluated for short quantum codes.