On evaluation codes coming from a tower of function fields

Carvalho, Cícero; Chara, María; Quoos, Luciane

doi:10.1016/j.jsc.2017.11.008

Cited by 4 publications

(4 citation statements)

References 7 publications

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“…4.1] a comparison between their results and ours can be taken as a direct measure of how well their method and our method compete. In Table 10: Estimated parameters of the codes C(δ) as found in the present paper [22,21,2] 8 [22,19,3] 8 [22,17,4] 8 [22,15,6] 8 [22,12,8] 8 [22,10,9] 8 [22,9,11] 8 [22,8,12] 8 [22,6,14] 8 [22,5,15] 8 [22,3,17] 8 [22,2,18] 8 [22,1,21] 8 [22,19,3] 8 [22,17,4] 8 [22,15,5] 8 …”

Section: Dual Affine Variety Codes From the Klein Quarticmentioning

confidence: 91%

“…3.2, Ex. 4.1] and in [10] which are among the seminal papers on Feng-Rao theory. One way of viewing our method is to consider it as a way of exhuming what (in our understanding) is the most basic principle that makes the Feng-Rao bound work, and to employ this principle in a novel manner.…”

Section: Introductionmentioning

confidence: 99%

“…To estimate the minimum distance of a primary affine variety code C(I, L) the most common approach (e.g. [21,13,4,3,12]) is to establish information on ≺ (F ) using only information on lm(F ) and paying in the analysis no attention to the coefficients of lower terms. I.e.…”

Section: Introductionmentioning

confidence: 99%

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On affine variety codes from the Klein quartic

Geil

Özbudak

2018

Cryptogr. Commun.

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We study a family of primary affine variety codes defined from the Klein quartic. The duals of these codes have previously been treated in [12, Ex. 3.2]. Among the codes that we construct almost all have parameters as good as the best known codes according to [9] and in the remaining few cases the parameters are almost as good. To establish the code parameters we apply the footprint bound [10,7] from Gröbner basis theory and for this purpose we develop a new method where we inspired by Buchbergers algorithm perform a series of symbolic computations.

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Section: Dual Affine Variety Codes From the Klein Quarticmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

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On affine variety codes from the Klein quartic

Geil

Özbudak

2018

Cryptogr. Commun.

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“…We define the footprint-bound of the code C A as the integer FB(C A ). This bound has been extensively studied in the literature, see for example [4,11,16] and the references therein.…”

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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From primary to dual affine variety codes over the Klein quartic

Geil

2022

Des. Codes Cryptogr.

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On evaluation codes coming from a tower of function fields

Cited by 4 publications

References 7 publications

On affine variety codes from the Klein quartic

On affine variety codes from the Klein quartic

On decoding hyperbolic codes

From primary to dual affine variety codes over the Klein quartic

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