Using commutative algebra methods we study the generalized minimum distance function (gmd functio... more 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.
In this paper we use polarization to study the behavior of the depth and regularity of a monomial... more In this paper we use polarization to study the behavior of the depth and regularity of a monomial ideal I, locally at a variable xi, when we lower the degree of all the highest powers of the variable xi occurring in the minimal generating set of I, and examine the depth and regularity of powers of edge ideals of clutters using combinatorial optimization techniques. If I is the edge ideal of an unmixed clutter with the max-flow min-cut property, we show that the powers of I have non-increasing depth and non-decreasing regularity. In particular edge ideals of unmixed bipartite graphs have non-decreasing regularity. We are able to show that the symbolic powers of the ideal of covers of the clique clutter of a strongly perfect graph have non-increasing depth. A similar result holds for the ideal of covers of a uniform ideal clutter.
Using commutative algebra methods we study the generalized minimum distance function (gmd functio... more 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.
In this paper we use polarization to study the behavior of the depth and regularity of a monomial... more In this paper we use polarization to study the behavior of the depth and regularity of a monomial ideal I, locally at a variable xi, when we lower the degree of all the highest powers of the variable xi occurring in the minimal generating set of I, and examine the depth and regularity of powers of edge ideals of clutters using combinatorial optimization techniques. If I is the edge ideal of an unmixed clutter with the max-flow min-cut property, we show that the powers of I have non-increasing depth and non-decreasing regularity. In particular edge ideals of unmixed bipartite graphs have non-decreasing regularity. We are able to show that the symbolic powers of the ideal of covers of the clique clutter of a strongly perfect graph have non-increasing depth. A similar result holds for the ideal of covers of a uniform ideal clutter.
Uploads
Papers by Carlos Vivares