57,114 research outputs found
Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids
To each linear code over a finite field we associate the matroid of its
parity check matrix. We show to what extent one can determine the generalized
Hamming weights of the code (or defined for a matroid in general) from various
sets of Betti numbers of Stanley-Reisner rings of simplicial complexes
associated to the matroid
A generalization of weight polynomials to matroids
Generalizing polynomials previously studied in the context of linear codes,
we define weight polynomials and an enumerator for a matroid . Our main
result is that these polynomials are determined by Betti numbers associated
with graded minimal free resolutions of the Stanley-Reisner ideals of and
so-called elongations of . Generalizing Greene's theorem from coding theory,
we show that the enumerator of a matroid is equivalent to its Tutte polynomial.Comment: 21 page
Higher weight spectra of Veronese codes
We study q-ary linear codes C obtained from Veronese surfaces over finite
fields. We show how one can find the higher weight spectra of these codes, or
equivalently, the weight distribution of all extension codes of C over all
field extensions of the field with q elements. Our methods will be a study of
the Stanley-Reisner rings of a series of matroids associated to each code CComment: 14 page
Stanley-Reisner resolution of constant weight linear codes
Given a constant weight linear code, we investigate its weight hierarchy and
the Stanley-Reisner resolution of its associated matroid regarded as a
simplicial complex. We also exhibit conditions on the higher weights sufficient
to conclude that the code is of constant weigh
The minimum distance of sets of points and the minimum socle degree
Let be a field of characteristic 0. Let be a reduced finite set of points, not all contained in a
hyperplane. Let be the maximum number of points of
contained in any hyperplane, and let . If
is the ideal of , then in \cite{t1}
it is shown that for , has a lower bound expressed in terms
of some shift in the graded minimal free resolution of . In these notes we
show that this behavior is true in general, for any : , where and is the last module in
the graded minimal free resolution of . In the end we also prove that this
bound is sharp for a whole class of examples due to Juan Migliore (\cite{m}).Comment: 11 page
Vanishing ideals over graphs and even cycles
Let X be an algebraic toric set in a projective space over a finite field. We
study the vanishing ideal, I(X), of X and show some useful degree bounds for a
minimal set of generators of I(X). We give an explicit description of a set of
generators of I(X), when X is the algebraic toric set associated to an even
cycle or to a connected bipartite graph with pairwise disjoint even cycles. In
this case, a fomula for the regularity of I(X) is given. We show an upper bound
for this invariant, when X is associated to a (not necessarily connected)
bipartite graph. The upper bound is sharp if the graph is connected. We are
able to show a formula for the length of the parameterized linear code
associated with any graph, in terms of the number of bipartite and
non-bipartite components
Complete intersections in binomial and lattice ideals
For the family of graded lattice ideals of dimension 1, we establish a
complete intersection criterion in algebraic and geometric terms. In positive
characteristic, it is shown that all ideals of this family are binomial set
theoretic complete intersections. In characteristic zero, we show that an
arbitrary lattice ideal which is a binomial set theoretic complete intersection
is a complete intersection.Comment: Internat. J. Algebra Comput., to appea
Magnitude cohomology
Magnitude homology was introduced by Hepworth and Willerton in the case of
graphs, and was later extended by Leinster and Shulman to metric spaces and
enriched categories. Here we introduce the dual theory, magnitude cohomology,
which we equip with the structure of an associative unital graded ring. Our
first main result is a 'recovery theorem' showing that the magnitude cohomology
ring of a finite metric space completely determines the space itself. The
magnitude cohomology ring is non-commutative in general, for example when
applied to finite metric spaces, but in some settings it is commutative, for
example when applied to ordinary categories. Our second main result explains
this situation by proving that the magnitude cohomology ring of an enriched
category is graded-commutative whenever the enriching category is cartesian. We
end the paper by giving complete computations of magnitude cohomology rings for
several large classes of graphs.Comment: 27 page
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