Generalized Sudan’s List Decoding for Order Domain Codes

Geil, Olav; Matsumoto, Ryutaroh

doi:10.1007/978-3-540-77224-8_9

Cited by 6 publications

(10 citation statements)

References 9 publications

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“…First to see that (X 3 , X) is OWB we must show that HX rem G = X 3 X rem G for all H with lm(H) = X 3 . We have (16) no matter what are a 1 , . .…”

Section: Example 3 Consider the Idealsmentioning

confidence: 99%

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Evaluation Codes from an Affine Variety Code Perspective

Geil

2008

Series on Coding Theory and Cryptology

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Evaluation codes (also called order domain codes) are traditionally introduced as generalized one-point geometric Goppa codes. In the present paper we will give a new point of view on evaluation codes by introducing them instead as particular nice examples of affine variety codes. Our study includes a reformulation of the usual methods to estimate the minimum distances of evaluation codes into the setting of affine variety codes. Finally we describe the connection to the theory of one-point geometric Goppa codes.

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“…First to see that (X 3 , X) is OWB we must show that HX rem G = X 3 X rem G for all H with lm(H) = X 3 . We have (16) no matter what are a 1 , . .…”

Section: Example 3 Consider the Idealsmentioning

confidence: 99%

“…Many papers deal with decoding of evaluation codes. Among these are [20], [6], and [16]. A study of the function µ on different families of semigroups Γ can be found in [5] and [34].…”

Section: Bibliographical Notesmentioning

confidence: 99%

Evaluation Codes from an Affine Variety Code Perspective

Geil

2008

Series on Coding Theory and Cryptology

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“…In combination with a preparation step this will allow us to correct more errors. The idea of a preparation step comes from [10]. The improved information regarding the zeros is derived by strengthening results reported by Dvir et al in [6].…”

Section: An Interpretation Of the Guruswami-sudan List Decoding Algormentioning

confidence: 90%

Weighted Reed–Muller codes revisited

Geil

Thomsen

2012

Des. Codes Cryptogr.

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We consider weighted Reed-Muller codes over point ensemble S1 × · · · × Sm where Si needs not be of the same size as Sj. For m = 2 we determine optimal weights and analyze in detail what is the impact of the ratio |S1|/|S2| on the minimum distance. In conclusion the weighted Reed-Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed-Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49,11,28] Joyner code.

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“…Moreover, in [19], the authors also present a generalization of the well-known list-decoding algorithm of Reed-Solomon codes given by Guruswami and Sudan [13] for the case of Reed-Muller codes. Then, Olav and Matsumoto [9], generalized this algorithm to evaluation codes from arbitrary order domains, in particular for hyperbolic codes. For more information about list decoding for AG codes see [2].…”

Section: Introductionmentioning

confidence: 99%

On decoding hyperbolic codes

Camps-Moreno¹,

García-Marco²,

López³

et al. 2021

Preprint

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Few decoding algorithms for hyperbolic codes are known in the literature, this article tries to fill this gap. The first part of this work compares hyperbolic codes and Reed-Muller codes. In particular, we determine when a Reed-Muller code is a hyperbolic code. As a byproduct, we state when a hyperbolic code has greater dimension than a Reed-Muller code when they both have the same minimum distance. We use the previous ideas to describe how to decode a hyperbolic code using the largest Reed-Muller code contained in it, or alternatively using the smallest Reed-Muller code that contains it. A combination of these two algorithms is proposed for the case when hyperbolic codes are defined by polynomials in two variables. Then, we compare hyperbolic codes and Cube codes (tensor product of Reed-Solomon codes) and we propose decoding algorithms of hyperbolic codes based on their closest Cube codes. Finally, we adapt to hyperbolic codes the Geil and Matsumoto's generalization of Sudan's list decoding algorithm.

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Generalized Sudan’s List Decoding for Order Domain Codes

Abstract: Abstract. We generalize Sudan's list decoding algorithm without multiplicity to evaluation codes coming from arbitrary order domains. The number of correctable errors by the proposed method is larger than the original list decoding without multiplicity.

Cited by 6 publications

References 9 publications

Evaluation Codes from an Affine Variety Code Perspective

Evaluation Codes from an Affine Variety Code Perspective

Weighted Reed–Muller codes revisited

On decoding hyperbolic codes

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