Generalized Hamming weights of affine Cartesian codes
Copyright policy )Generalized Hamming weights of affine Cartesian codes
In this article, we give the answer to the following question: Given a field $\mathbb{F}$, finite subsets $A_1,\dots,A_m$ of $\mathbb{F}$, and $r$ linearly independent polynomials $f_1,\dots,f_r \in \mathbb{F}[x_1,\dots,x_m]$ of total degree at most $d$. What is the maximal number of common zeros $f_1,\dots,f_r$ can have in $A_1 \times \cdots \times A_m$? For $\mathbb{F}=\mathbb{F}_q$, 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.
12 Pages
- Technical University of Denmark Denmark
Finite element method, Linearly independents, affine Hilbert functions, Zero-dimensional, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry, Cartesians, Combinatorics of partially ordered sets, Mathematics - Algebraic Geometry, Numerical Methods, FOS: Mathematics, Algebraic Geometry (math.AG), Data Processing, Codes (symbols), Affine Hilbert functions, Zero dimensional varieties, affine Cartesian codes, Algebra, Generalized Hamming weight, Affine Cartesian codes, Generalized Hamming weights, Finite fields, zero dimensional varieties, generalized Hamming weights, Finite subsets, Hilbert functions, Geometric methods (including applications of algebraic geometry) applied to coding theory
Finite element method, Linearly independents, affine Hilbert functions, Zero-dimensional, Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry, Cartesians, Combinatorics of partially ordered sets, Mathematics - Algebraic Geometry, Numerical Methods, FOS: Mathematics, Algebraic Geometry (math.AG), Data Processing, Codes (symbols), Affine Hilbert functions, Zero dimensional varieties, affine Cartesian codes, Algebra, Generalized Hamming weight, Affine Cartesian codes, Generalized Hamming weights, Finite fields, zero dimensional varieties, generalized Hamming weights, Finite subsets, Hilbert functions, Geometric methods (including applications of algebraic geometry) applied to coding theory
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