134 research outputs found
A Polymatroid Approach to Generalized Weights of Rank Metric Codes
We consider the notion of a -polymatroid, due to Shiromoto, and the
more general notion of -demi-polymatroid, and show how generalized
weights can be defined for them. Further, we establish a duality for these
weights analogous to Wei duality for generalized Hamming weights of linear
codes. The corresponding results of Ravagnani for Delsarte rank metric codes,
and Martinez-Penas and Matsumoto for relative generalized rank weights are
derived as a consequence.Comment: 22 pages; with minor revisions in the previous versio
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
Almost affine codes and matroids
In this thesis we study various types of block codes, like linear, mutlti-linear, almost affine codes. We also look at how these codes can be described by associated matroids. In addition we look at flags (chains) of codes and see how their behavior can be described using demi-matroids. We also introduce weight polynomials for almost affine codes
Coxeter submodular functions and deformations of Coxeter permutahedra
We describe the cone of deformations of a Coxeter permutahedron, or
equivalently, the nef cone of the toric variety associated to a Coxeter
complex. This family of polytopes contains polyhedral models for the
Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and
associahedra. Our description extends the known correspondence between
generalized permutahedra, polymatroids, and submodular functions to any finite
reflection group.Comment: Minor edits. To appear in Advances of Mathematic
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