Decoding Hermitian Codes with Sudan’s Algorithm

Høholdt, Tom; Nielsen, R. Refslund

doi:10.1007/3-540-46796-3_26

Cited by 37 publications

(12 citation statements)

References 6 publications

(11 reference statements)

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“…This problem has been considered in the literature, with significant success. In particular, it is now known how to implement the interpolation step in O(n 2 ) time, when the output list size is a constant [29,31]. Similar running times are also known for the root finding problem (which suffices for the second step in the algorithms above) [2,11,28,29,31,39].…”

Section: Factor Q and Report All Polynomialsmentioning

confidence: 97%

“…In particular, it is now known how to implement the interpolation step in O(n 2 ) time, when the output list size is a constant [29,31]. Similar running times are also known for the root finding problem (which suffices for the second step in the algorithms above) [2,11,28,29,31,39]. Together these algorithms lead to the possibility that a good implementation of list-decoding may actually even be able to compete with the classical Berkelkamp-Massey decoding algorithm in terms of efficiency.…”

Section: Factor Q and Report All Polynomialsmentioning

confidence: 99%

“…E.g., in the case of algebraic-geometry codes, the issue becomes that of how the codes are specified. For most well-known families of such codes, the standard specifications do lead to polynomial time solutions [11,28,29,39]. For arbitrary codes, however it is a priori unclear if a natural representation could lead to a polynomial time decoding algorithm.…”

Section: Redundant Residue Number System Codesmentioning

confidence: 99%

“…In fact in the absence of a complete characterization of all algebraic geometry codes, it is unclear as to what is a natural representation for all of them. [19] suggest a potential representation that is reasonably succinct (polynomial sized in the generator matrix), that allows this and other necessary tasks to be solved in polynomial time, by using the algorithms of [11] and [29].…”

Section: Redundant Residue Number System Codesmentioning

confidence: 99%

See 3 more Smart Citations

Ideal Error-Correcting Codes: Unifying Algebraic and Number-Theoretic Algorithms

Sudan¹

2001

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

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Abstract. Over the past five years a number of algorithms decoding some well-studied error-correcting codes far beyond their "error-correcting radii" have been developed. These algorithms, usually termed as listdecoding algorithms, originated with a list-decoder for Reed-Solomon codes [36,17], and were soon extended to decoders for Algebraic Geometry codes [33,17] and as also some number-theoretic codes [12,6,16]. In addition to their enhanced decoding capability, these algorithms enjoy the benefit of being conceptually simple, fairly general [16], and are capable of exploiting soft-decision information in algebraic decoding [24]. This article surveys these algorithms and highlights some of these features.

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Section: Factor Q and Report All Polynomialsmentioning

confidence: 97%

Section: Factor Q and Report All Polynomialsmentioning

confidence: 99%

Section: Redundant Residue Number System Codesmentioning

confidence: 99%

Section: Redundant Residue Number System Codesmentioning

confidence: 99%

See 2 more Smart Citations

Ideal Error-Correcting Codes: Unifying Algebraic and Number-Theoretic Algorithms

Sudan¹

2001

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

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“…Furthermore, we will design a novel O(n 7/3 ) algorithm for the linear algebra step of list decoding of certain algebraic-geometric-codes from plane curves of block length n with lists of length . We remark that, using other means, Høholdt and Refslund Nielsen [18] have obtained an algorithm for list decoding on Hermitian curves which solves both steps of the algorithm in [15] more efficiently.…”

Section: )mentioning

confidence: 99%

A displacement approach to decoding algebraic codes

Olshevsky¹,

Shokrollahi²

2003

Fast Algorithms for Structured Matrices: Theory and Applications

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Abstract. Algebraic coding theory is one of the areas that routinely gives rise to computational problems involving various structured matrices, such as Hankel, Vandermonde, Cauchy matrices, and certain generalizations thereof. Their structure has often been used to derive efficient algorithms; however, the use of the structure was pattern-specific, without applying a unified technique. In contrast, in several other areas, where structured matrices are also widely encountered, the concept of displacement rank was found to be useful to derive efficient algorithms in a unified manner (i.e., not depending on a particular pattern of structure). The latter technique allows one to "compress," in a unified way, different types of n × n structured matrices to only αn parameters. This typically leads to computational savings (in many applications the number α, called the displacement rank, is a small fixed constant).In this paper we demonstrate the power of the displacement structure approach by deriving in a unified way efficient algorithms for a number of decoding problems. We accelerate the Sudan's list decoding algorithm for ReedSolomon codes, its generalization to algebraic-geometric codes by Shokrollahi and Wasserman, and the improvement of Guruswami and Sudan in the case of Reed-Solomon codes. In particular, we notice that matrices that occur in the context of list decoding have low displacement rank, and use this fact to derive algorithms that use O(n 2 ) and O(n 7/3 ) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively. Here denotes the length of the list; assuming that is constant, this gives algorithms of running time O(n 2 ) and O(n 7/3 ), which is the same as the running time of conventional decoding algorithms. We also present efficient parallel algorithms for the above tasks.To the best of our knowledge this is the first application of the concept of displacement rank to the unified derivation of several decoding algorithms; the technique can be useful in finding efficient and fast methods for solving other decoding problems.

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Basic Principles of Non‐Binary Codes

Carrasco

Johnston

2008

Non‐Binary Error Control Coding for Wireless Communication and Data Storage

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Decoding Hermitian Codes with Sudan’s Algorithm

Cited by 37 publications

References 6 publications

Ideal Error-Correcting Codes: Unifying Algebraic and Number-Theoretic Algorithms

Ideal Error-Correcting Codes: Unifying Algebraic and Number-Theoretic Algorithms

A displacement approach to decoding algebraic codes

Basic Principles of Non‐Binary Codes

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