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Added by
Rafael Villarreal H
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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No Arabic abstract
We compute the basic parameters (dimension, length, minimum distance) of affine evaluation codes defined on a cartesian product of finite sets. Given a sequence of positive integers, we construct an evaluation code, over a degenerate torus, with prescribed parameters. As an application of our results, we recover the formulas for the minimum distance of various families of evaluation codes.
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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 $G$.
Let K be a finite field and let X* be an affine algebraic toric set parameterized by monomials. We give an algebraic method, using Groebner bases, to compute the length and the dimension of C_X*(d), the parameterized affine code of degree d on the set X*. If Y is the projective closure of X*, it is shown that C_X^*(d) has the same basic parameters that C_Y(d), the parameterized projective code on the set Y. If X* is an affine torus, we compute the basic parameters of C_X*(d). We show how to compute the vanishing ideals of X* and Y.
We show that the degree of a graded lattice ideal of dimension 1 is the order of the torsion subgroup of the quotient group of the lattice. This gives an efficient method to compute the degree of this type of lattice ideals.
In this paper we introduce a new type of code, called projective nested cartesian code. It is obtained by the evaluation of homogeneous polynomials of a fixed degree on a certain subset of $mathbb{P}^n(mathbb{F}_q)$, and they may be seen as a generalization of the so-called projective Reed-Muller codes. We calculate the length and the dimension of such codes, a lower bound for the minimum distance and the exact minimum distance in a special case (which includes the projective Reed-Muller codes). At the end we show some relations between the parameters of these codes and those of the affine cartesian codes.
Let $mathbb{F}_{q}$ denote the finite field of order $q,$ let $m_1,m_2,cdots,m_{ell}$ be positive integers satisfying $gcd(m_i,q)=1$ for $1 leq i leq ell,$ and let $n=m_1+m_2+cdots+m_{ell}.$ Let $Lambda=(lambda_1,lambda_2,cdots,lambda_{ell})$ be fixed, where $lambda_1,lambda_2,cdots,lambda_{ell}$ are non-zero elements of $mathbb{F}_{q}.$ In this paper, we study the algebraic structure of $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ and their dual codes with respect to the standard inner product on $mathbb{F}_{q}^n.$ We provide necessary and sufficient conditions for the existence of a self-dual $Lambda$-multi-twisted code of length $n$ over $mathbb{F}_{q},$ and obtain enumeration formulae for all self-dual and self-orthogonal $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}.$ We also derive some sufficient conditions under which a $Lambda$-multi-twisted code is LCD. We determine the parity-check polynomial of all $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ and obtain a BCH type bound on their minimum Hamming distances. We also determine generating sets of dual codes of some $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ from the generating sets of the codes. Besides this, we provide a trace description for all $Lambda$-multi-twisted codes of length $n$ over $mathbb{F}_{q}$ by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure.
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