On the next-to-minimal weight of affine cartesian codes

Carvalho, Cícero; Neumann, Víctor

doi:10.1016/j.ffa.2016.11.005

Cited by 15 publications

(12 citation statements)

References 14 publications

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“…The codes obtained in this way are called affine cartesian codes. Affine cartesian codes were defined in [10] and further studied in, for example, [1,12,8,2]. In [10,Theorem 3.8] the authors determined the minimum distance of these codes.…”

Section: Affine Cartesian Codes and Their Higher Weightsmentioning

confidence: 99%

Generalized Hamming weights of affine Cartesian codes

Beelen

Datta

2018

Finite Fields and Their Applications

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In this article, we give the answer to the following question: Given a field F, finite subsets A 1 , . . . , Am of F, and r linearly independent polynomials f 1 , . . . , fr ∈ F[x 1 , . . . , xm] of total degree at most d. What is the maximal number of common zeros f 1 , . . . , fr can have in A 1 ×· · ·×Am? For F = Fq, the finite field with q elements, answering this question is equivalent to determining the generalized Hamming weights of the so-called affine Cartesian codes. Seen in this light, our work is a generalization of the work of Heijnen-Pellikaan for Reed-Muller codes to the significantly larger class of affine Cartesian codes.

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Section: Affine Cartesian Codes and Their Higher Weightsmentioning

confidence: 99%

Generalized Hamming weights of affine Cartesian codes

Beelen

Datta

2018

Finite Fields and Their Applications

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“…In [11], the author shows some results on higher Hamming weights of Cartesian codes and gives a different proof for the minimum distance using the concepts of Gröbner basis and footprint of an ideal. In [12] the authors find several values for the second least weight of codewords, also known as the next-to-minimal Hamming weight. In [2] the authors find the generalized Hamming weights and the dual of Cartesian codes.…”

Section: Introductionmentioning

confidence: 99%

Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes

López

Matthews

Soprunov

2020

Des. Codes Cryptogr.

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A monomial-Cartesian code is an evaluation code defined by evaluating a set of monomials over a Cartesian product. It is a generalization of some families of codes in the literature, for instance toric codes, affine Cartesian codes and J -affine variety codes. In this work we use the vanishing ideal of the Cartesian product to give a description of the dual of a monomial-Cartesian code. Then we use such description of the dual to prove the existence of quantum error correcting codes and MDS quantum error correcting codes. Finally we show that the direct product of monomial-Cartesian codes is a locally recoverable code with t-availability if at least t of the components are locally recoverable codes.

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“…This generalizes the classical work [15] of Heijnen and Pelikaan towards the determination of all the generalized Hamming weights of the Reed-Muller codes. Several articles, for example [3,4], are devoted towards the determination of the next to minimal weights of affine Cartesian codes.…”

Section: Introductionmentioning

confidence: 99%

Relative generalized Hamming weights of affine Cartesian codes

Datta

2020

Des. Codes Cryptogr.

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We explicitly determine all the relative generalized Hamming weights of affine Cartesian codes using the notion of footprints and results from extremal combinatorics. This generalizes the previous works on the determination of relative generalized Hamming weights of Reed-Muller codes by Geil and Martin, as well as the determination of all the generalized Hamming weights of the affine Cartesian codes by Beelen and Datta. The author is supported by a postdoctoral fellowship from DST-RCN grant INT/NOR/RCN/ICT/P-03/2018.

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On the next-to-minimal weight of affine cartesian codes

Cited by 15 publications

References 14 publications

Generalized Hamming weights of affine Cartesian codes

Generalized Hamming weights of affine Cartesian codes

Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes

Relative generalized Hamming weights of affine Cartesian codes

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