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Abstmct-We reveal an equivalence relation between the construction of a new class of low density MDS array codes, that we call B-Code, and a com-binatorial problem known as perfect one-factorization of complete graphs. We use known perfect one-factors of complete graphs to create constructions and decoding algorithms for both B-Code and its dual code.(More)
For a nite multigraph G, the reliability function of G is the probability R G (q) that if each edge of G is deleted independently with probability q then the remaining edges of G induce a connected spanning subgraph of G; this is a polynomial function of q. In 1992, Brown and Colbourn conjectured that for any connected multigraph G, if q 2 C is such that R(More)
The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial k m k (G)y k has only real nonpositive zeros, in which m k (G) is the number of k-edge matchings of a graph G. There is a stronger multivariate version of this theorem. We provide a general method by which " theorems of Heilmann-Lieb type " can be proved for a wide(More)
The generalized theta graph s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : :; s k 1. We prove that the roots of the chromatic polynomial ((s 1 ;:::;s k ; z) of a k-ary generalized theta graph all lie in the disc jz ? 1j 1 + o(1)] k= logk, uniformly in the path lengths s i. Moreover, we prove that(More)