Support Constrained Generator Matrices of Gabidulin Codes in Characteristic Zero

Gabidulin codes over fields of characteristic zero were recently constructed by Augot et al., whenever the Galois group of the underlying field extension is cyclic. In parallel, the interest in sparse generator matrices of Reed–Solomon and Gabidulin codes has increased lately, due to applications in distributed computations. In particular, a certain condition pertaining to the intersection of zero entries at different rows, was shown to be necessary and sufficient for the existence of the sparsest possible generator matrix of Gabidulin codes over finite fields. In this paper we complete the picture by showing that the same condition is also necessary and sufficient for Gabidulin codes over fields of characteristic zero.Our proof builds upon and extends tools from the finite-field case, combines them with a variant of the Schwartz–Zippel lemma over automorphisms, and provides a simple randomized construction algorithm whose probability of success can be arbitrarily close to one. In addition, potential applications for low-rank matrix recovery are discussed

Un teorema sobre desigualdades rango lineales que dependen de la caractertística del cuerpo finito

A linear rank inequality is a linear inequality that holds by dimensions of vector spaces over any finite field. A characteristic-dependent linear rank inequality is also a linear inequality that involves dimensions of vector spaces but this holds over finite fields of determined characteristics, and does not in general hold over other characteristics. In this paper, using as guide binary matrices whose ranks depend on the finite field where they are defined, we show a theorem which explicitly produces characteristic-dependent linear rank inequalities; this theorem generalizes results previously obtained in the literature.Una desigualdad rango lineal es una desigualdad lineal que es válida para dimensiones de espacios vectoriales sobre un cuerpo finito. Una desigualdad rango lineal dependiente de la característica es también una desigualdad lineal para dimensiones de espacios vectoriales pero ésta es válida sobre cuerpos finitos de determinada característica, y no es válida en general sobre otras características. En este documento, usando como guía matrices binarias cuyos rangos dependen del cuerpo finito en donde están definidas, nosotros presentamos un teorema que produce explícitamente desigualdades rango lineales dependientes de la característica; ´este teorema generaliza resultados obtenidos previamente en la literatura

A Dual Fano, and Dual Non-Fano Matroidal Network

Matroidal networks are useful tools in furthering research in network coding. They have been used to show the limitations of linear coding solutions. In this paper we examine the basic information on network coding and matroid theory. We then go over the method of creating matroidal networks. Finally we construct matroidal networks from the dual of the fano matroid and the dual of the non-fano matroid, and breifly discuss some coding solutions