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

Greedy weights for matroids

We introduce greedy weights of matroids, inspired by those for linear codes. We show that a Wei duality holds for two of these types of greedy weights for matroids. Moreover we show that in the cases where the matroids involved are associated to linear codes, our definitions coincide with those for codes. Thus our Wei duality is a generalization of that for linear codes given by Schaathun. In the last part of the paper we show how some important chains of cycles of the matroids appearing, correspond to chains of component maps of minimal resolutions of the independence complex of the corresponding matroids. We also relate properties of these resolutions to chainedness and greedy weights of the matroids, and in many cases codes, that appear.Comment: 17 page

Free Resolutions Associated to Representable Matroids

As a matroid is naturally a simplicial complex, one can study its combinatorial properties via the associated Stanley-Reisner ideal and its corresponding free resolution. Using results by Johnsen and Verdure, we prove that a matroid is the dual to a perfect matroid design if and only if its corresponding Stanley-Reisner ideal has a pure free resolution, and, motivated by applications to their generalized Hamming weights, characterize free resolutions corresponding to the vector matroids of the parity check matrices of Reed-Solomon codes and certain BCH codes. Furthermore, using an inductive mapping cone argument, we construct a cellular resolution for the matroid duals to finite projective geometries and discuss consequences for finite affine geometries. Finally, we provide algorithms for computing such cellular resolutions explicitly

Aplicaciones a la teoría de códigos de los números de Betti de una matroide

En teoría de la información uno de los problemas que existe es el detectar y corregir los errores en un mensaje enviado, donde el mensaje es enviado a través de un canal de información. Los códigos que más se estudian son los códigos lineales C, donde C es un subespacio lineal de Fnq, con Fq un campo finito con q elementos y n ∈ N. Un invariante numérico de suma importancia para C (ver [17], [27]) es la mínima distancia de Hamming d(C) que proporciona una solución al problema de detección y corrección de errores. En 1991, V.K. Wei generaliza el peso de Hamming y define para cada i ∈ {1, . . . , k} con k la dimensión del código, el i-ésimo peso generalizado de Hamming di(C), donde d1(C) = d(C) y 1 ≤ d1(C) < d2(C) < · · · < dk(C). Dichos pesos tienen relevancia por su estrecha relación con los problemas de códigos conocidos como wire-tap channel of type II ([25]) y las funciones t-resilientes ([8])