Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields
Copyright policy )Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields
Let K be a finite field with q elements and let X be a subset of a projective space P^{s-1}, over the field K, which is parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper are in determining the structure of I(X) and some of their invariants. It is shown that I(X) is a lattice ideal. We introduce the notion of a parameterized code arising from X and present algebraic methods to compute and study its dimension, length and minimum distance. For a parameterized code arising from a connected graph we are able to compute its length and to make our results more precise. If the graph is non-bipartite, we show an upper bound for the minimum distance. We also study the underlying geometric structure of X.
Finite Fields Appl., to appear
- UNIVERSIDADE DE SAO PAULO Brazil
- Instituto Politécnico Nacional Mexico
- Universidade Federal de Pernambuco Brazil
Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry, Commutative Algebra (math.AC), Index of regularity, parameterized codes, Theoretical Computer Science, Finite ground fields in algebraic geometry, Mathematics - Algebraic Geometry, Binomial and lattice ideals, evaluation codes, Projective variety, binomial and lattice ideals, Degree, FOS: Mathematics, Minimum distance, Algebraic Geometry (math.AG), Engineering(all), Linear codes (general theory), Algebra and Number Theory, Applied Mathematics, 13P25, 94B27, Mathematics - Commutative Algebra, Parameters of a code, Evaluation codes, Hilbert function, Gröbner bases, Parameterized codes, Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Geometric methods (including applications of algebraic geometry) applied to coding theory
Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry, Commutative Algebra (math.AC), Index of regularity, parameterized codes, Theoretical Computer Science, Finite ground fields in algebraic geometry, Mathematics - Algebraic Geometry, Binomial and lattice ideals, evaluation codes, Projective variety, binomial and lattice ideals, Degree, FOS: Mathematics, Minimum distance, Algebraic Geometry (math.AG), Engineering(all), Linear codes (general theory), Algebra and Number Theory, Applied Mathematics, 13P25, 94B27, Mathematics - Commutative Algebra, Parameters of a code, Evaluation codes, Hilbert function, Gröbner bases, Parameterized codes, Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Geometric methods (including applications of algebraic geometry) applied to coding theory
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