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Optimizing Cauchy ReedSolomon codes for faulttolerant network storage applications
 In NCA06: 5th IEEE International Symposium on Network Computing Applications
, 2006
"... NOTE: NCA’s page limit is rather severe: 8 pages. As a result, the final paper is pretty much a hatchet job of the original submission. I would recommend reading the technical report version of this paper, because it presents the material with some accompanying tutorial material, and is easier to re ..."
Abstract

Cited by 28 (11 self)
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NOTE: NCA’s page limit is rather severe: 8 pages. As a result, the final paper is pretty much a hatchet job of the original submission. I would recommend reading the technical report version of this paper, because it presents the material with some accompanying tutorial material, and is easier to read. The technical report is available at:
Coding over an Erasure Channel with a Large Alphabet Size
, 2008
"... An erasure channel with a fixed alphabet size q, where q ≫ 1, is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming maximum likelihood decoding). Assuming a memoryless erasure channel ..."
Abstract

Cited by 5 (2 self)
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An erasure channel with a fixed alphabet size q, where q ≫ 1, is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming maximum likelihood decoding). Assuming a memoryless erasure channel, the error exponent of MDS codes are compared with that of random codes. It is shown that the envelopes of these two exponents are identical for rates above the critical rate. Noting the optimality of MDS codes, it is concluded that random coding is exponentially optimal as long as the block size N satisfies N < q + 1.
Path Diversity in Packet Switched Networks: Performance Analysis and Rate Allocation
"... Path diversity works by setting up multiple parallel connections between the end points using the topological path redundancy of the network. In this paper, Forward Error Correction (FEC) is applied across multiple independent paths to enhance the endtoend reliability. Internet paths are modeled ..."
Abstract

Cited by 5 (3 self)
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Path diversity works by setting up multiple parallel connections between the end points using the topological path redundancy of the network. In this paper, Forward Error Correction (FEC) is applied across multiple independent paths to enhance the endtoend reliability. Internet paths are modeled as erasure GilbertElliot channels [1], [2]. First, it is shown that over any erasure channel, Maximum Distance Separable (MDS) codes achieve the minimum probability of irrecoverable loss among all block codes of the same size. Then, we prove the probability of irrecoverable loss decays exponentially for the asymptotically large number of paths. Moreover, it is shown that in the optimal rate allocation, each path is assigned a positive rate iff its quality is above a certain threshold. The quality of a path is defined as the percentage of the time it spends in the bad state. Finally, using dynamic programming, a heuristic suboptimal algorithm with polynomial runtime is proposed for rate allocation over the available paths. This algorithm converges to the asymptotically optimal rate allocation when the number of paths is large. The simulation results show that the proposed algorithm approximates the optimal rate allocation very closely, and provides significant performance improvement compared to the alternative schemes of rate allocation.
Path Diversity over the Internet: Performance Analysis and Rate Allocation
"... Path diversity works by setting up multiple parallel connections between the end points using the topological path redundancy of the network. In this paper, Forward Error Correction (FEC) is applied across multiple independent paths to enhance the endtoend reliability. Internet paths are modeled ..."
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Cited by 1 (1 self)
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Path diversity works by setting up multiple parallel connections between the end points using the topological path redundancy of the network. In this paper, Forward Error Correction (FEC) is applied across multiple independent paths to enhance the endtoend reliability. Internet paths are modeled as erasure GilbertElliot channels [1], [2]. First, it is shown that over any erasure channel, Maximum Distance Separable (MDS) codes achieve the minimum probability of irrecoverable loss among all block codes of the same size. Then, we prove the probability of irrecoverable loss decays exponentially for the asymptotically large number of paths. Moreover, it is shown that in the optimal rate allocation, each path is assigned a positive rate iff its quality is above a certain threshold. The quality of a path is defined as the percentage of the time it spends in the bad state. Finally, using dynamic programming, a heuristic suboptimal algorithm with polynomial runtime is proposed for rate allocation over the available paths. This algorithm converges to the asymptotically optimal rate allocation when the number of paths is large. The simulation results show that the proposed algorithm approximates the optimal rate allocation very closely, and provides significant performance improvement compared to the alternative schemes of rate allocation.
Path Diversity over Packet Switched Networks: Performance Analysis and Rate Allocation
, 2008
"... Path diversity works by setting up multiple parallel connections between the end points using the topological path redundancy of the network. In this paper, Forward Error Correction (FEC) is applied across multiple independent paths to enhance the endtoend reliability. Network paths are modeled as ..."
Abstract

Cited by 1 (0 self)
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Path diversity works by setting up multiple parallel connections between the end points using the topological path redundancy of the network. In this paper, Forward Error Correction (FEC) is applied across multiple independent paths to enhance the endtoend reliability. Network paths are modeled as erasure GilbertElliot channels [1]– [5]. It is known that over any erasure channel, Maximum Distance Separable (MDS) codes achieve the minimum probability of irrecoverable loss among all block codes of the same size [6], [7]. Based on the adopted model for the error behavior, we prove that the probability of irrecoverable loss for MDS codes decays exponentially for an asymptotically large number of paths. Then, optimal rate allocation problem is solved for the asymptotic case where the number of paths is large. Moreover, it is shown that in such asymptotically optimal rate allocation, each path is assigned a positive rate iff its quality is above a certain threshold. The quality of a path is defined as the percentage of the time it spends in the bad state. Finally, using dynamic programming, a heuristic suboptimal algorithm with polynomial runtime is proposed for rate allocation over a finite number of paths. This algorithm converges to the asymptotically optimal rate allocation when the number of paths is large. The simulation results show that the proposed algorithm approximates the optimal rate allocation (found by exhaustive search) very closely for practical number of paths, and provides significant performance improvement compared to the alternative schemes of rate allocation.
1 Alphabet Size
"... An erasure channel with a fixed alphabet size q, where q ≫ 1, is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming maximum likelihood decoding). Assuming a memoryless erasure channel ..."
Abstract
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An erasure channel with a fixed alphabet size q, where q ≫ 1, is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming maximum likelihood decoding). Assuming a memoryless erasure channel, the error exponent of MDS codes are compared with that of random codes. It is shown that the envelopes of these two exponents are identical for rates above the critical rate. Noting the optimality of MDS codes, it is concluded that random coding is exponentially optimal as long as the block size N satisfies N < q + 1. 1 I.
1 the Internet using MDS Codes
"... Path diversity works by setting up multiple parallel connections between the end points using the topological path redundancy of the network. In this paper, Forward Error Correction (FEC) is applied across multiple independent paths to enhance the endtoend reliability. Internet paths are modeled as ..."
Abstract
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Path diversity works by setting up multiple parallel connections between the end points using the topological path redundancy of the network. In this paper, Forward Error Correction (FEC) is applied across multiple independent paths to enhance the endtoend reliability. Internet paths are modeled as erasure GilbertElliot channels [1], [2]. First, it is shown that over any erasure channel, Maximum Distance Separable (MDS) codes achieve the minimum probability of irrecoverable loss among all block codes of the same size. Then, based on the adopted model for Internet paths, we prove that the probability of irrecoverable loss for MDS codes decays exponentially for the asymptotically large number of paths. Moreover, it is shown that in the optimal rate allocation, each path is assigned a positive rate iff its quality is above a certain threshold. The quality of a path is defined as the percentage of the time it spends in the bad state. In other words, including a redundant path improves the reliability iff this condition is satisfied. Finally, using dynamic programming, a heuristic suboptimal algorithm with polynomial runtime is proposed for rate allocation over the available paths. This algorithm converges to the asymptotically optimal rate allocation when the number of paths is large. The simulation results show that the proposed algorithm approximates the optimal rate allocation very closely for the practical number of paths, and provides significant performance improvement compared to the alternative schemes of rate allocation. 1 I.
Optimal Coding for the Erasure Channel with Arbitrary Alphabet Size
"... An erasure channel with an arbitrary alphabet size q is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming maximum likelihood decoding). Next, based on the performance of MDS codes, a ..."
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An erasure channel with an arbitrary alphabet size q is studied. It is proved that over any erasure channel (with or without memory), Maximum Distance Separable (MDS) codes achieve the minimum probability of error (assuming maximum likelihood decoding). Next, based on the performance of MDS codes, a lowerbound on the error probability of block codes over the erasure channel is derived. It is shown that this lowerbound (denoted by Lm(N, K, q)) is valid whether or not an MDS code of size [N, K] exists. For the case of a memoryless erasure channel with any arbitrary alphabet size, the exponential behavior of the lowerbound is studied. Finally, it is proved that both random codes and random linear codes have the same exponent as Lm(N, K, q) for the memoryless erasure channel of any arbitrary alphabet size and rates above the critical rate. In other words, considering rates above the critical rate, both random codes and random linear codes are exponentially optimal over the memoryless erasure channel.
FNTbased ReedSolomon Erasure Codes
, 907
"... Abstract—This paper presents a new construction of MaximumDistance Separable (MDS) ReedSolomon erasure codes based on Fermat Number Transform (FNT). Thanks to FNT, these codes support practical coding and decoding algorithms with complexity O(n log n), where n is the number of symbols of a codewor ..."
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Abstract—This paper presents a new construction of MaximumDistance Separable (MDS) ReedSolomon erasure codes based on Fermat Number Transform (FNT). Thanks to FNT, these codes support practical coding and decoding algorithms with complexity O(n log n), where n is the number of symbols of a codeword. An opensource implementation shows that the encoding speed can reach 150Mbps for codes of length up to several 10,000s of symbols. These codes can be used as the basic component of the Information Dispersal Algorithm (IDA) system used in a several P2P systems. I. INTRODUCTION AND RELATED WORK Erasure and network coding concepts were recently used in several kinds of networks in order to improve the throughput and the reliability of the systems. In distributed storage systems like P2P or RAIDbased systems,