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Register a new userThe second generalized Hamming weight of some evaluation codes arising from a projective torus
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Added by
Manuel Gonz\\'alez Sarabia
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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In this paper we find the second generalized Hamming weight of some evaluation codes arising from a projective torus, and it allows to compute the second generalized Hamming weight of the codes parameterized by the edges of any complete bipartite graph. Also, at the beginning, we obtain some results about the generalized Hamming weights of some evaluation codes arising from a complete intersection when the minimum distance is known and they are non--degenerate codes. Finally we give an example where we use these results to determine the complete weight hierarchy of some codes.
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We study affine cartesian codes, which are a Reed-Muller type of evaluation codes, where polynomials are evaluated at the cartesian product of n subsets of a finite field F_q. These codes appeared recently in a work by H. Lopez, C. Renteria-Marquez and R. Villareal and, in a generalized form, in a work by O. Geil and C. Thomsen. Using methods from Grobner basis theory we determine the second Hamming weight (also called next-to-minimal weight) for particular cases of affine cartesian codes and also some higher Hamming weights of this type of code.
Let $G$ be a connected graph and let $mathbb{X}$ be the set of projective points defined by the column vectors of the incidence matrix of $G$ over a field $K$ of any characteristic. We determine the generalized Hamming weights of the Reed--Muller-type code over the set $mathbb{X}$ in terms of graph theoretic invariants. As an application to coding theory we show that if $G$ is non-bipartite and $K$ is a finite field of ${rm char}(K) eq 2$, then the $r$-th generalized Hamming weight of the linear code generated by the rows of the incidence matrix of $G$ is the $r$-th weak edge biparticity of $G$. If ${rm char}(K)=2$ or $G$ is bipartite, we prove that the $r$-th generalized Hamming weight of that code is the $r$-th edge connectivity of $G$.
Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code ${cal C}$, identify for each dimension $ u$, the smallest size of the support of a subcode of ${cal C}$ of dimension $ u$. The GHW of a code are of interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use in bounding the state complexity of the trellis representation of the code. In prior work by the same authors, a code-shortening algorithm was employed to derive upper bounds on the GHW of binary projective, Reed-Muller (PRM) codes. In the present paper, we derive a matching lower bound by adapting the proof techniques used originally for Reed-Muller (RM) codes by Wei. This results in a characterization of the GHW hierarchy of binary PRM codes.
In this paper we will estimate the main parameters of some evaluation codes which are known as projective parameterized codes. We will find the length of these codes and we will give a formula for the dimension in terms of the Hilbert function associated to two ideals, one of them being the vanishing ideal of the projective torus. Also we will find an upper bound for the minimum distance and, in some cases, we will give some lower bounds for the regularity index and the minimum distance. These lower bounds work in several cases, particularly for any projective parameterized code associated to the incidence matrix of uniform clutters and then they work in the case of graphs.
The generalized Hamming weights of a linear code have been extensively studied since Wei first use them to characterize the cryptography performance of a linear code over the wire-tap channel of type II. In this paper, we investigate the generalized Hamming weights of three classes of linear codes constructed through defining sets and determine them partly for some cases. Particularly, in the semiprimitive case we solve an problem left in Yang et al. (IEEE Trans. Inf. Theory 61(9): 4905--4913, 2015).
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