Comments on and Corrections to “On the Equivalence of Generalized Concatenated Codes and Generalized Error Location Codes” [Mar 00 642-649]

Fan, Jihao; Chen, Hanwu

doi:10.1109/tit.2017.2689782

Cited by 4 publications

(5 citation statements)

References 2 publications

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“…According to [24], [32], if we do not transpose the constituent companion matrices in (2), we can obtain another representation of the parity check matrix H T as follows…”

Section: B Classical Tensor Product Codesmentioning

confidence: 99%

Asymmetric Quantum Concatenated and Tensor Product Codes With Large Z-Distances

Fan

Wang

et al. 2021

IEEE Trans. Commun.

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In many quantum channels, dephasing errors occur more frequently than the amplitude errorsa phenomenon that has been exploited for performance gains and other benefits through asymmetric quantum codes (AQCs). In this paper, we present a new construction of AQCs by combining classical concatenated codes (CCs) with tensor product codes (TPCs), called asymmetric quantum concatenated and tensor product codes (AQCTPCs) which have the following three advantages. First, only the outer codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes, and any classical linear code can be used for the inner, which makes AQCTPCs very easy to construct. Second, most AQCTPCs are highly degenerate, which means they can correct many more errors than their classical TPC counterparts. Consequently, we construct several families of AQCs with better parameters than known results in the literature. Especially, we derive a first family of binary AQCs with the Zdistance larger than half the block length. Third, AQCTPCs can be efficiently decoded although they

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“…According to [24], [32], if we do not transpose the constituent companion matrices in (2), we can obtain another representation of the parity check matrix H T as follows…”

Section: B Classical Tensor Product Codesmentioning

confidence: 99%

Asymmetric Quantum Concatenated and Tensor Product Codes With Large Z-Distances

Fan

Wang

et al. 2021

IEEE Trans. Commun.

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“…In this section we give a concatenation construction of long QBECCs from two short component codes based on the quantum tensor product codes structure [7], [25].…”

Section: B Concatenation Construction Of Qbeccs Based On Quantum Tens...mentioning

confidence: 99%

Construction and Performance of Quantum Burst Error Correction Codes for Correlated Errors

Fan

Hsieh

Chen

et al. 2018

2018 IEEE International Symposium on Information Theory (ISIT)

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In practical communication and computation systems, errors occur predominantly in adjacent positions rather than in a random manner. In this paper, we develop a stabilizer formalism for quantum burst error correction codes (QBECC) to combat such error patterns in the quantum regime. Our contributions are as follows. Firstly, we derive an upper bound for the correctable burst errors of QBECCs, the quantum Reiger bound (QRB). This bound generalizes the quantum Singleton bound for standard quantum error correction codes (QECCs). Secondly, we propose two constructions of QBECCs: one by heuristic computer search and the other by concatenating two quantum tensor product codes (QTPCs). We obtain several new QBECCs with better parameters than existing codes with the same coding length. Moreover, some of the constructed codes can saturate the quantum Reiger bounds. Finally, we perform numerical experiments for our constructed codes over Markovian correlated depolarizing quantum memory channels, and show that QBECCs indeed outperform standard QECCs in this scenario.

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“…If quantum code Q is pure and code D is linear over GF (4), then we get a Hermitian dual-containing code C = D ⊥ h which has the same minimum distance with Q. All the component codes [11,6,5] and [12,6,6] in Table 3 and Table 4 are trace-Hermitian dual-containing additive codes that correspond to BKQCs Q in [15] and the MAGMA database. By using MAGMA, we know that they happen to be linear and Hermitian dual-containing codes over GF (4), and have the same minimum distance with the corresponding BKQCs Q.…”

Section: Qtpcs With Component Codes Derived From Online Code Tables A...mentioning

confidence: 99%

“…The tensor product concatenating scheme could significantly improve the efficiency of the inner parity code while retaining a similar performance. In [37,3,19,11], generalized TPCs (also called generalized error location codes) were shown to be equivalent to generalized concatenated codes.…”

Section: Introductionmentioning

confidence: 99%

“…Let C L be a TPC which has the same parameters with C by replacing the component matrix H c1 in H [C] with LH c1 . Denote by H [CL] the parity check matrix of C L and we use(11) with untransposed companion matrices as the parity check matrix of C L . Then C and C L have the same parameters and error control abilities but have different parity check matrix structures.…”

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confidence: 99%

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On Quantum Tensor Product Codes

Fan¹,

Li²,

Hsieh³

et al. 2016

Preprint

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We present a general framework for the construction of quantum tensor product codes (QTPC). In a classical tensor product code (TPC), its parity check matrix is constructed via the tensor product of parity check matrices of the two component codes. We show that by adding some constraints on the component codes, several classes of dual-containing TPCs can be obtained. By selecting different types of component codes, the proposed method enables the construction of a large family of QTPCs and they can provide a wide variety of quantum error control abilities. In particular, if one of the component codes is selected as a burst-error-correction code, then QTPCs have quantum multiple-burst-error-correction abilities, provided these bursts fall in distinct subblocks. Compared with concatenated quantum codes (CQC), the component code selections of QTPCs are much more flexible than those of CQCs since only one of the component codes of QTPCs needs to satisfy the dual-containing restriction. We show that it is possible to construct QTPCs with parameters better than other classes of quantum error-correction codes (QECC), e.g., CQCs and quantum BCH codes. Many QTPCs are obtained with parameters better than previously known quantum codes available in the literature. Several classes of QTPCs that can correct multiple quantum bursts of errors are constructed based on reversible cyclic codes and maximum-distance-separable (MDS) codes.

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Comments on and Corrections to “On the Equivalence of Generalized Concatenated Codes and Generalized Error Location Codes” [Mar 00 642-649]

Cited by 4 publications

References 2 publications

Asymmetric Quantum Concatenated and Tensor Product Codes With Large Z-Distances

Asymmetric Quantum Concatenated and Tensor Product Codes With Large Z-Distances

Construction and Performance of Quantum Burst Error Correction Codes for Correlated Errors

On Quantum Tensor Product Codes

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