Generalized minimum distance functions

Using commutative algebra methods, we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a system of homogeneous polynomials has in any given finite set...

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Bibliographic Details
Published in:Journal of algebraic combinatorics 2019-11, Vol.50 (3), p.317-346
Main Authors: González-Sarabia, Manuel, Martínez-Bernal, José, Villarreal, Rafael H., Vivares, Carlos E.
Format: Article
Language:English
Subjects:
Cartesian coordinates
Codes
Combinatorial analysis
Combinatorics
Computer Science
Convex and Discrete Geometry
Fields (mathematics)
Footprints
Group Theory and Generalizations
Lattices
Lower bounds
Mathematical analysis
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
Polynomials
Rings (mathematics)
Weight
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Summary:Using commutative algebra methods, we study the generalized minimum distance function (gmd function) and the corresponding generalized footprint function of a graded ideal in a polynomial ring over a field. The number of solutions that a system of homogeneous polynomials has in any given finite set of projective points is expressed as the degree of a graded ideal. If X is a set of projective points over a finite field and I is its vanishing ideal, we show that the gmd function and the Vasconcelos function of I are equal to the r th generalized Hamming weight of the corresponding Reed–Muller-type code C X ( d ) of degree d . We show that the generalized footprint function of I is a lower bound for the r th generalized Hamming weight of C X ( d ) . Then, we present some applications to projective nested Cartesian codes. To give applications of our lower bound to algebraic coding theory, we show an interesting integer inequality. Then, we show an explicit formula and a combinatorial formula for the second generalized Hamming weight of an affine Cartesian code.
ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-018-0855-x