Relative Generalized Hamming Weights of One-Point Algebraic Geometric Codes

Geil, Olav; Martin, Stefano; Matsumoto, Ryutaroh; Ruano, Diego; Luo, Yuan

doi:10.1109/tit.2014.2345375

Cited by 34 publications

(21 citation statements)

References 40 publications

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“…The parameterized code of order d associated to the toric set X, or the code of order d parameterized by the monomials (1), is the image of the evaluation map ev d , and it is denoted by C X (d). Therefore…”

Section: Preliminariesmentioning

confidence: 99%

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Some bounds for the relative generalized Hamming weights of some evaluation codes

Sarabia

2016

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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In this paper we find some bounds for the relative generalized Hamming weights of some codes parameterized by a set of monomials of the same degree. Also we compare the relative generalized Hamming weights of the codes CX (d) and C X (d) when X is embedded in X. We use these results to obtain some lower bounds for the relative generalized Hamming weights of the codes parameterized by the edges of any connected bipartite graph with bipartition (V1, V2) and where |V1| = n1, |V2| = n2, in terms of the relative generalized Hamming weights of the codes associated to the projective tori Tn 1 −1 and Tn 2 −1.

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Section: Preliminariesmentioning

confidence: 99%

“…Moreover some equivalences, inequalities and bounds are given in [20]. Recently the behavior of the RGHW of one point algebraic geometric codes is analyzed in [1]. In the case of Hermitian codes, the RGHW are often much larger than the corresponding generalized Hamming weights.…”

Section: Introductionmentioning

confidence: 99%

Some bounds for the relative generalized Hamming weights of some evaluation codes

Sarabia

2016

Analele Universitatii "Ovidius" Constanta - Seria Matematica

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“…We here briefly review the connections, revealed first in [11] and studied in [12] with one-point AG codes, to emphasize the significance of relative generalized Hamming weights for secret sharing schemes.…”

Section: Bounds For Relative Generalized Hamming Weightsmentioning

confidence: 99%

“…Then it was shown in [11,12] that relative generalized Hamming weights precisely measure the information leakage in the secret sharing scheme based on linear codes. Recently Geil et al [12] extended the order bound for the relative generalized Hamming weights of one-point evaluation AG codes and their dual codes.…”

Section: Introductionmentioning

confidence: 99%

Bounds for generalized Hamming weights of general AG codes

Lee

2015

Finite Fields and Their Applications

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We present a good bound for the generalized Hamming weights of multi-point evaluation and differential AG codes. It is a natural generalization of the order bound for onepoint AG codes. As an example, we determine the third generalized Hamming weight of one particular two-point evaluation Hermitian code. Then we extend the bound for the relative generalized Hamming weights, which are essential measures of the performance of secret sharing schemes based on linear codes. The access structure of one particular secret sharing scheme based on the previous two-point Hermitian code and its subcode is analyzed using the new bound.

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“…Determining the RGHWs of a code with respect to its subcodes is an interesting research work [4,5]. In order to determine part of the RGHWs of C with respect to a k 1 -dimensional subcode C 1 , the second author gave an alternative definition of the RGHWs as follows [5].…”

Section: Introductionmentioning

confidence: 99%

On the second relative greedy weight

Liu

2015

Cryptogr. Commun.

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Based on the wire-tap channel of type II with two users given by Luo et al. in 2005, we introduce the relative greedy weights. A finite projective geometry method is presented in order to describe the relative greedy weights. By using this finite projective geometry method, the optimal 3-dimensional q-ary codes are searched so as to maximize the effort for the adversary to obtain the second data symbol on condition that he has obtained the first data symbol from the wire-tap channel of type II with two users.

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Relative Generalized Hamming Weights of One-Point Algebraic Geometric Codes

Cited by 34 publications

References 40 publications

Some bounds for the relative generalized Hamming weights of some evaluation codes

Some bounds for the relative generalized Hamming weights of some evaluation codes

Bounds for generalized Hamming weights of general AG codes

On the second relative greedy weight

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