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Shutao Xia - One of the best experts on this subject based on the ideXlab platform.
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bounds and constructions of locally repairable codes parity Check Matrix approach
IEEE Transactions on Information Theory, 2020Co-Authors: Jie Hao, Shutao Xia, Kenneth W Shum, Bin Chen, Yixian YangAbstract:A locally repairable code (LRC) is a linear code such that every code symbol can be recovered by accessing a small number of other code symbols. In this paper, we study bounds and constructions of LRCs from the viewpoint of parity-Check matrices. Firstly, a simple and unified framework based on parity-Check Matrix to analyze the bounds of LRCs is proposed, and several new explicit bounds on the minimum distance of LRCs in terms of the field size are presented. In particular, we give an alternate proof of the Singleton-like bound for LRCs first proved by Gopalan et al. Some structural properties on optimal LRCs that achieve the Singleton-like bound are given. Then, we focus on constructions of optimal LRCs over the binary field. It is proved that there are only five classes of possible parameters with which optimal binary LRCs exist. Moreover, by employing the proposed parity-Check Matrix approach, we completely enumerate all these five classes of optimal binary LRCs attaining the Singleton-like bound in the sense of equivalence of linear codes.
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Bounds and Constructions of Locally Repairable Codes: Parity-Check Matrix Approach
arXiv: Information Theory, 2016Co-Authors: Jie Hao, Shutao XiaAbstract:A code symbol of a linear code is said to have locality r if this symbol could be recovered by at most r other code symbols. An (n,k,r) locally repairable code (LRC) with all symbol locality is a linear code with length n, dimension k, and locality r for all symbols. Recently, there are lots of studies on the bounds and constructions of LRCs, most of which are essentially based on the generator Matrix of the linear code. Up to now, the most important bounds of minimum distance for LRCs might be the well-known Singleton-like bound and the Cadambe-Mazumdar bound concerning the field size. In this paper, we study the bounds and constructions of LRCs from views of parity-Check matrices. Firstly, we set up a new characterization of the parity-Check Matrix for an LRC. Then, the proposed parity-Check Matrix is employed to analyze the minimum distance. We give an alternative simple proof of the well-known Singleton-like bound for LRCs with all symbol locality, and then easily generalize it to a more general bound, which essentially coincides with the Cadambe-Mazumdar bound and includes the Singleton-like bound as a specific case. Based on the proposed characterization of parity-Check matrices, necessary conditions of meeting the Singleton-like bound are obtained, which naturally lead to a construction framework of good LRCs. Finally, two classes of optimal LRCs based on linearized polynomial theories and Vandermonde matrices are obtained under the construction framework.
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Stopping Set Distributions of Some Reed–Muller Codes
IEEE Transactions on Information Theory, 2011Co-Authors: Yong Jiang, Shutao XiaAbstract:Stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let C be a binary [n,k] linear code with parity-Check Matrix H, where the rows of H may be dependent. A stopping set S of C with parity-Check Matrix H is a subset of column indices of H such that the restriction of H to S does not contain a row of weight one. The stopping set distribution {Ti(H)}i=0n enumerates the number of stopping sets with size i of C with parity-Check Matrix H. Note that stopping sets and stopping set distribution are related to the parity-Check Matrix H of C. Let H* be the parity-Check Matrix of C which is formed by all the nonzero codewords of its dual code C⊥. A parity-Check Matrix H is called BEC-optimal if Ti(H)=Ti(H*), i=0,1,..., n and H has the smallest number of rows. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-Check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-Check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes, and the extended Hamming codes, which are some Reed-Muller codes or their shortening or puncturing versions.
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Stopping Set Distributions of Some Linear Codes
arXiv: Information Theory, 2010Co-Authors: Yong Jiang, Shutao XiaAbstract:Stopping sets and stopping set distribution of an low-density parity-Check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let $C$ be a binary $[n,k]$ linear code with parity-Check Matrix $H$, where the rows of $H$ may be dependent. A stopping set $S$ of $C$ with parity-Check Matrix $H$ is a subset of column indices of $H$ such that the restriction of $H$ to $S$ does not contain a row of weight one. The stopping set distribution $\{T_i(H)\}_{i=0}^n$ enumerates the number of stopping sets with size $i$ of $C$ with parity-Check Matrix $H$. Note that stopping sets and stopping set distribution are related to the parity-Check Matrix $H$ of $C$. Let $H^{*}$ be the parity-Check Matrix of $C$ which is formed by all the non-zero codewords of its dual code $C^{\perp}$. A parity-Check Matrix $H$ is called BEC-optimal if $T_i(H)=T_i(H^*), i=0,1,..., n$ and $H$ has the smallest number of rows. On the BEC, iterative decoder of $C$ with BEC-optimal parity-Check Matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-Check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-Check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes and the extended Hamming codes.
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Stopping Set Distributions of Some Linear Codes
2006 IEEE Information Theory Workshop - ITW '06 Chengdu, 2006Co-Authors: Shutao XiaAbstract:In this paper, the stopping set distributions (SSD) of some well-known binary linear codes are determined by using finite geometry theory. Similar to the weight distribution of a binary linear code, the SSD {Ti(H)}n i=0 enumerates the number of stopping sets with size i of a linear code with parity-Check Matrix H. First, we deal with the simplex codes and Hamming codes. With parity-Check Matrix formed by all the weight 3 codewords of the Hamming code, the SSD of the simplex code is completely determined with explicit formula. With parity-Check Matrix formed by all the nonzero codewords of the simplex code, the SSD of the Hamming code is completely determined with two recursive equations. Then, the first order Reed-Muller codes and the extended Hamming codes are discussed. With parity-Check Matrix formed by all the weight 4 codewords of the extended Hamming code, the SSD of the first order Reed-Muller code is completely determined with explicit formula. With parity-Check Matrix formed by all the minimum codewords of the first order Reed-Muller code, the SSD of the extended Hamming code is completely determined with two recursive equations
Giovanni Cancellieri - One of the best experts on this subject based on the ideXlab platform.
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Strict-Sense Time-Invariant Convolutional Codes in Their Parity Check Matrix
Polynomial Theory of Error Correcting Codes, 2014Co-Authors: Giovanni CancellieriAbstract:The syndrome former sub-Matrix is introduced. The generator Matrix of a convolutional code can be constructed by means of a procedure called null ex-OR sum of clusters of syndromes. Inversely, it is possible to operate by means of the column construction of the parity Check Matrix. The importance of a parity Check Matrix in minimal form is stressed. A strict-sense time-invariant convolutional code in its parity Check Matrix is characterized by a unique interleaved parity Check polynomial. It is possible to distinguish between low-rate and high-rate convolutional codes, taking into account their parity Check Matrix. The dual of a low-rate convolutional code is a high-rate convolutional code, but the outermost parts of the frame show different structures in the two matrices. A systematic encoder circuit based on the unique interleaved parity Check polynomial is presented. Traditional encoder circuits for convolutional codes described by means of the parity Check Matrix are discussed, considering a unique shift register in observer arrangement. Not well designed convolutional codes and the presence of periodic rows in their parity Check Matrix are studied. The tail-biting arrangement of a convolutional code is revisited looking at its parity Check Matrix. When the code is not well-designed, the tail-biting convolutional code has a generator Matrix not of full rank, and its periodic row in the parity Check Matrix vanishes. A second conceptual bridge between cyclic block codes and convolutional codes is presented, now based on the parity Check Matrix. Modified lengthening in the generator Matrix and H-extension in the parity Check Matrix support this correspondence. A family tree for many classes of error correcting codes is depicted. Not well-designed convolutional codes can be considered the dual with respect to convolutional codes whose parity Check Matrix is not in its minimal form.
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Parity Check Matrix Approach to Linear Block Codes
Polynomial Theory of Error Correcting Codes, 2014Co-Authors: Giovanni CancellieriAbstract:The parity Check Matrix can be assumed for an alternative description of a linear code. The relationship with the generator Matrix is not univocal, except when such matrices are systematic. Upper triangular generator matrices and lower triangular parity Check matrices are presented, showing the advantage of a clear recognition of information and control symbol positions. Error correction is outlined by means of properly processing single-error syndromes. A strict-sense time-invariant code in its parity Check Matrix is characterized by a unique parity Check polynomial. The concept of dual code is discussed. Code puncturation and code shortening are interpreted as dual operations. Constant-length puncturation and constant-length shortening are described. Periodic parity Check polynomials in lengthened cyclic codes are constructed. The parity Check Matrix of a modified lengthened cyclic code is derived. The difference between periodic and non-periodic parity Check rows is stressed. H-extended cyclic codes and modified H-extended cyclic codes are introduced. Generalized repetition codes represent dual codes of generalized parity Check codes. The whole family of Hamming codes is constructed by repeated lengthening operations. The whole family of simplex codes is constructed by repeated H-extensions. Direct product codes are described by means of their parity Check Matrix. An encoder circuit based on the parity Check polynomial is depicted. Its state diagram is studied, and the trellis obtained from it is compared with the traditional generator trellis. Finally some considerations about the parity Check Matrix of nonbinary block codes are developed, with particular attention to their error correction capability.
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Wide-Sense Time-Invariant Convolutional Codes in Their Parity Check Matrix
Polynomial Theory of Error Correcting Codes, 2014Co-Authors: Giovanni CancellieriAbstract:The circuit for syndrome calculation in a convolutional code is introduced. The most general method for obtaining the parity Check Matrix in a convolutional code, starting from its generator Matrix, is presented. The minimal encoder circuit is discussed. Wide-sense time-invariant convolutional codes form a closed ensemble. Code puncturation is studied for convolutional codes looking at their parity Check Matrix. Tail-biting w.s. time-invariant convolutional codes are examined, considering both the generator and the parity Check Matrix. Unwrapping a quasi-cyclic code for obtaining convolutional codes, in general w.s. time-invariant, leads to a second conceptual bridge between these two classes of codes, in particular regarding their parity Check Matrix. This observation is important especially for low-density parity-Check codes (LDPC codes). Array codes are now interpreted as LDPC block codes without short cycles of 1-symbols. It is possible to obtain an unwrapped form for such codes, in order to construct interesting LDPC convolutional codes. The equivalence between an array code with two component codes and a direct product code between two parity Check codes is demonstrated. The concept of truncated circulants is transferred to a description based on the parity Check Matrix.
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Wide-Sense Time-Invariant Block Codes in Their Parity Check Matrix
Polynomial Theory of Error Correcting Codes, 2014Co-Authors: Giovanni CancellieriAbstract:A wide-sense time-invariant block code in its parity Check Matrix exhibits more than one parity Check polynomial periodically repeating. Quasi-cyclic codes are revisited from this point of view. Reordered forms of quasi-cyclic codes are treated. Shortening or lengthening, with respect to the quasi-cyclic condition, are possible here only with steps of integer periods. The concept of code duality is extended. Code puncturation is described for quasi-cyclic codes. Constant-length shortening and constant-length puncturation are dual operations by which an information symbol is transformed into a control symbols and vice versa. Modified lengthening and modified H-extension are discussed also for quasi-cyclic codes. Encoder circuits, state diagrams and trellises are finally outlined.
Sayyed Rasoul Mousavi - One of the best experts on this subject based on the ideXlab platform.
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Stopping Set Elimination by Parity-Check Matrix Extension via Integer Linear Programming
IEEE Transactions on Communications, 2015Co-Authors: Hossein Falsafain, Sayyed Rasoul MousaviAbstract:Error-rate floor phenomenon is known to be a serious impediment to the use of low-density parity-Check (LDPC) codes for some practical applications that demand high data reliability. In the case of binary erasure channels (BECs), certain error-prone patterns, known as stopping sets, are proven to cause this performance degradation. A possible approach to diminish this drawback over BECs is to eliminate stopping sets by parity-Check Matrix extension. Given a parity-Check Matrix $H$ , and a list $\mathcal{L} $ of its stopping sets, we present an integer linear programming (ILP) formulation to find a parity-Check equation which eliminates the maximum number of stopping sets in $\mathcal{L} $ . One of the distinguishing advantages of the proposed scheme is its flexibility for modifications such as: limiting the weight of the new parity-Check row, making the new row redundant or linearly independent, 4-cycle avoidance, and taking into account the sizes of stopping sets. Armed with these adjustments, the method can provide good performance improvements, as evidenced by simulation results. Furthermore, for a given $\varrho\in\mathbb{N} $ , by extending the basic formulation, we provide an ILP formulation for finding a set of size $\varrho$ of parity-Check equations which can best eliminate the stopping sets in $\mathcal{L}$ , among all such sets.
Yong Jiang - One of the best experts on this subject based on the ideXlab platform.
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Stopping Set Distributions of Some Reed–Muller Codes
IEEE Transactions on Information Theory, 2011Co-Authors: Yong Jiang, Shutao XiaAbstract:Stopping sets and stopping set distribution of a linear code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let C be a binary [n,k] linear code with parity-Check Matrix H, where the rows of H may be dependent. A stopping set S of C with parity-Check Matrix H is a subset of column indices of H such that the restriction of H to S does not contain a row of weight one. The stopping set distribution {Ti(H)}i=0n enumerates the number of stopping sets with size i of C with parity-Check Matrix H. Note that stopping sets and stopping set distribution are related to the parity-Check Matrix H of C. Let H* be the parity-Check Matrix of C which is formed by all the nonzero codewords of its dual code C⊥. A parity-Check Matrix H is called BEC-optimal if Ti(H)=Ti(H*), i=0,1,..., n and H has the smallest number of rows. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-Check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-Check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes, and the extended Hamming codes, which are some Reed-Muller codes or their shortening or puncturing versions.
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Stopping Set Distributions of Some Linear Codes
arXiv: Information Theory, 2010Co-Authors: Yong Jiang, Shutao XiaAbstract:Stopping sets and stopping set distribution of an low-density parity-Check code are used to determine the performance of this code under iterative decoding over a binary erasure channel (BEC). Let $C$ be a binary $[n,k]$ linear code with parity-Check Matrix $H$, where the rows of $H$ may be dependent. A stopping set $S$ of $C$ with parity-Check Matrix $H$ is a subset of column indices of $H$ such that the restriction of $H$ to $S$ does not contain a row of weight one. The stopping set distribution $\{T_i(H)\}_{i=0}^n$ enumerates the number of stopping sets with size $i$ of $C$ with parity-Check Matrix $H$. Note that stopping sets and stopping set distribution are related to the parity-Check Matrix $H$ of $C$. Let $H^{*}$ be the parity-Check Matrix of $C$ which is formed by all the non-zero codewords of its dual code $C^{\perp}$. A parity-Check Matrix $H$ is called BEC-optimal if $T_i(H)=T_i(H^*), i=0,1,..., n$ and $H$ has the smallest number of rows. On the BEC, iterative decoder of $C$ with BEC-optimal parity-Check Matrix is an optimal decoder with much lower decoding complexity than the exhaustive decoder. In this paper, we study stopping sets, stopping set distributions and BEC-optimal parity-Check matrices of binary linear codes. Using finite geometry in combinatorics, we obtain BEC-optimal parity-Check matrices and then determine the stopping set distributions for the Simplex codes, the Hamming codes, the first order Reed-Muller codes and the extended Hamming codes.
Minyi Guo - One of the best experts on this subject based on the ideXlab platform.
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optimizing the parity Check Matrix for efficient decoding of rs based cloud storage systems
International Parallel and Distributed Processing Symposium, 2019Co-Authors: Xin Xie, Han Qiu, Minyi Guo, Yuanyuan Dong, Yafei ZhaoAbstract:In large scale distributed systems such as cloud storage systems, erasure coding is a fundamental technique to provide high reliability at low monetary cost. Compared with the traditional disk arrays, cloud storage systems use an erasure coding scheme with both flexible fault tolerance and high scalability. Thus, Reed-Solomon (RS) Codes or RS-based codes are popular choices for cloud storage systems. However, the decoding performance for RS-based codes is not as good as XOR-based codes, which are optimized via investigating the relationships among different parity chains or reducing the computational complexity of Matrix multiplications. Therefore, exploring an efficient decoding method is highly desired. To address the above problem, in this paper, we propose an Advanced Parity-Check Matrix (APCM) based approach, which is extended from the original Parity-Check Matrix based (PCM) approach. Instead of improving the decoding performance of XOR-based codes in PCM, APCM focuses on optimizing the decoding efficiency for RS-based codes. Furthermore, APCM avoids the Matrix inversion computations and reduces the computational complexity of the decoding process. To demonstrate the effectiveness of the APCM, we conduct intensive experiments by using both RS-based and XOR-based codes under cloud storage environment. The results show that, compared to typical decoding methods, APCM improves the decoding speed by up to 32.31% in the Alibaba cloud storage system.
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IPDPS - Optimizing the Parity Check Matrix for Efficient Decoding of RS-Based Cloud Storage Systems
2019 IEEE International Parallel and Distributed Processing Symposium (IPDPS), 2019Co-Authors: Xin Xie, Han Qiu, Minyi Guo, Yuanyuan Dong, Yafei ZhaoAbstract:In large scale distributed systems such as cloud storage systems, erasure coding is a fundamental technique to provide high reliability at low monetary cost. Compared with the traditional disk arrays, cloud storage systems use an erasure coding scheme with both flexible fault tolerance and high scalability. Thus, Reed-Solomon (RS) Codes or RS-based codes are popular choices for cloud storage systems. However, the decoding performance for RS-based codes is not as good as XOR-based codes, which are optimized via investigating the relationships among different parity chains or reducing the computational complexity of Matrix multiplications. Therefore, exploring an efficient decoding method is highly desired. To address the above problem, in this paper, we propose an Advanced Parity-Check Matrix (APCM) based approach, which is extended from the original Parity-Check Matrix based (PCM) approach. Instead of improving the decoding performance of XOR-based codes in PCM, APCM focuses on optimizing the decoding efficiency for RS-based codes. Furthermore, APCM avoids the Matrix inversion computations and reduces the computational complexity of the decoding process. To demonstrate the effectiveness of the APCM, we conduct intensive experiments by using both RS-based and XOR-based codes under cloud storage environment. The results show that, compared to typical decoding methods, APCM improves the decoding speed by up to 32.31% in the Alibaba cloud storage system.
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SRDS - PCM: A Parity-Check Matrix Based Approach to Improve Decoding Performance of XOR-based Erasure Codes
2015 IEEE 34th Symposium on Reliable Distributed Systems (SRDS), 2015Co-Authors: Yongzhe Zhang, Minyi GuoAbstract:In large storage systems, erasure codes is a primary technique to provide high reliability with low monetary cost. Among various erasure codes, a major category called XORbased codes uses purely XOR operations to generate redundant data and offer low computational complexity. These codes are conventionally implemented via Matrix based method or several specialized non-Matrix based methods. However, these approaches are insufficient on decoding performance, which affects the reliability and availability of storage systems. To address the problem, in this paper, we propose a novel Parity-Check Matrix based (PCM) approach, which is a general-purpose method to implement XOR-based codes, and increases the decoding performance by using smaller and sparser matrices. To demonstrate the effectiveness of PCM, we conduct several experiments by using different XOR-based codes. The evaluation results show that, compared to typical Matrix based decoding methods, PCM can improve the decoding speed by up to a factor of 1.5× when using EVENODD code (an erasure code for correcting double disk failures), and accelerate the decoding process of STAR code (an erasure code for correcting triple disk failures) by up to a factor of 2.4×.