A method for solving key equation for decoding goppa codes

Sugiyama, Y.; Kasahara, Masao; Hirasawa, Shigeichi; Namekawa, T.

doi:10.1016/s0019-9958(75)90090-x

Cited by 325 publications

(182 citation statements)

References 3 publications

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“…From the sequence S (1) we construct three sequences of length N min by shifting, as follows: These reconstructed syndromes can be applied as the input to FT's Euclidean algorithm. Here, the intermediate results according to this algorithm are listed in Table I.…”

Section: <2>mentioning

confidence: 99%

“…The proof was given by Sugiyama, Kasahara, Hirasawa and Namekawa (SKHN, [1]) in 1975 and the equivalence to the Berlekamp-Massey (BM, [2]) algorithm, synthesizing the shortest linear feedback shiftregister (LFSR) generating one sequence S, is widely accepted. Feng and Tzeng (FT) generalized the SKHN approach in 1989 [3] and the BM algorithm in 1991 [4] to multiple sequences.…”

Section: Introductionmentioning

confidence: 99%

“…Feng and Tzeng (FT) generalized the SKHN approach in 1989 [3] and the BM algorithm in 1991 [4] to multiple sequences. Both generalizations synthesize the shortest linear feedback shift-register generating of s sequences S (0) , S (1) , . .…”

Section: Introductionmentioning

confidence: 99%

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Decoding Reed-Solomon codes up to the Sudan radius with the Euclidean algorithm

Zeh¹,

Li²

2010

2010 International Symposium on Information Theory &Amp; Its Applications

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Abstract-We modify the Euclidean algorithm of Feng and Tzeng to decode Reed-Solomon (RS) codes up to the Sudan radius. The basic steps are the virtual extension to an Interleaved RS code and the reformulation of the multi-sequence shiftregister problem of varying length to a multi-sequence problem of equal length. We prove the reformulation and analyze the complexity of our new decoding approach. Furthermore, the extended key equation, that describes the multi-sequence problem, is derived in an alternative polynomial way.

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Section: <2>mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Decoding Reed-Solomon codes up to the Sudan radius with the Euclidean algorithm

Zeh¹,

Li²

2010

2010 International Symposium on Information Theory &Amp; Its Applications

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“…Another generalizes the theory of Padé approximation (Beckermann & Labahn 1994;Forney, Jr. 1975;Giorgi et al 2003;Van Barel & Bultheel 1992). The interpretation of the Berlekamp/Massey algorithm as a specialization of the extended Euclidean algorithm (Dornstetter 1987;Sugiyama et al 1975) can be carried over to matrix polynomials (Coppersmith 1994;Thomé 2002) (see also Section 3 below). All approaches solve the classical Levinson-Durbin problem, which for matrix sequences becomes a block Toeplitz linear system (Kaltofen 1995).…”

Section: As In the Unblockedmentioning

confidence: 99%

On the complexity of computing determinants

Kaltofen

Villard²

2005

comput. complex.

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Abstract. We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n × n matrix A with integer entries in (n 3.2 log A ) 1+o(1) and (n 2.697263 log A ) 1+o(1) bit operations; here A denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors C 1 (log n) C 2 (loglog A ) C 3 for positive real constants C 1 , C 2 , C 3 . The bit complexity (n 3.2 log A ) 1+o(1) results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n 3.2+o(1) and O(n 2.697263 ) ring additions, subtractions and multiplications.

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“…The hard-decision decoding (HDD) algorithm is capable of successfully decoding any vector with no more than d min /2 errors, where d min is the minimum distance of the code. Even decades after its development, variants of Berlekamp's original formulation (Berlekamp-Massey (B-M) Algorithm [2], Euclid Algorithm [3]) are, overwhelmingly, the most implemented RS decoding procedures in modern communications.…”

Section: Introductionmentioning

confidence: 99%

A Low-Complexity Method for Chase-Type Decoding of Reed-Solomon Codes

Bellorado¹,

Kavcic

2006

2006 IEEE International Symposium on Information Theory

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Abstract-In this work we present a low-complexity implementation of Chase-type decoding of Reed-Solomon Codes. In such, we first use the soft-information available at the channel output to construct a test-set of 2 η vectors, equivalent in all except the η << n least reliable coordinate positions. We then give an interpolation procedure to construct a set of 2 η bivariate polynomials, with the roots of each specified by its corresponding test-vector. Here, test-vector similarity is exploited to share much of the required computation. Finally, we obtain the candidate message from the single z-linear factor of each bivariate polynomial. Although we provide an expression for the direct computation of each candidate message, the complexity of repeating this computation for each interpolation polynomial is prohibitive. We, thus, also present a reduced-complexity factorization (RCF) method to select a single polynomial that, with high probability, contains the correctly decoded message in its z-linear factor. Although suboptimal, the loss in performance of RCF decreases rapidly with increasing code length. We provide extensive simulation results showing that a significant performance increase over traditional hard-decision decoding is achievable with a comparable computational complexity (as implemented with the BerlekampMassey Algorithm).

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A method for solving key equation for decoding goppa codes

Cited by 325 publications

References 3 publications

Decoding Reed-Solomon codes up to the Sudan radius with the Euclidean algorithm

Decoding Reed-Solomon codes up to the Sudan radius with the Euclidean algorithm

On the complexity of computing determinants

A Low-Complexity Method for Chase-Type Decoding of Reed-Solomon Codes

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