Linear Fractional Network Coding and Representable Discrete Polymatroids

A linear Fractional Network Coding (FNC) solution over $\mathbb{F}_q$ is a linear network coding solution over $\mathbb{F}_q$ in which the message dimensions need not necessarily be the same and need not be the same as the edge vector dimension. Scalar linear network coding, vector linear network coding are special cases of linear FNC. In this paper, we establish the connection between the existence of a linear FNC solution for a network over $\mathbb{F}_q$ and the representability over $\mathbb{F}_q$ of discrete polymatroids, which are the multi-set analogue of matroids. All previously known results on the connection between the scalar and vector linear solvability of networks and representations of matroids and discrete polymatroids follow as special cases. An algorithm is provided to construct networks which admit FNC solution over $\mathbb{F}_q,$ from discrete polymatroids representable over $\mathbb{F}_q.$ Example networks constructed from discrete polymatroids using the algorithm are provided, which do not admit any scalar and vector solution, and for which FNC solutions with the message dimensions being different provide a larger throughput than FNC solutions with the message dimensions being equal.Comment: 8 pages, 5 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:1301.300

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 Linear Network Code Construction for General Integer Connections Based on the Constraint Satisfaction Problem

The problem of finding network codes for general connections is inherently difficult in capacity constrained networks. Resource minimization for general connections with network coding is further complicated. Existing methods for identifying solutions mainly rely on highly restricted classes of network codes, and are almost all centralized. In this paper, we introduce linear network mixing coefficients for code constructions of general connections that generalize random linear network coding (RLNC) for multicast connections. For such code constructions, we pose the problem of cost minimization for the subgraph involved in the coding solution and relate this minimization to a path-based Constraint Satisfaction Problem (CSP) and an edge-based CSP. While CSPs are NP-complete in general, we present a path-based probabilistic distributed algorithm and an edge-based probabilistic distributed algorithm with almost sure convergence in finite time by applying Communication Free Learning (CFL). Our approach allows fairly general coding across flows, guarantees no greater cost than routing, and shows a possible distributed implementation. Numerical results illustrate the performance improvement of our approach over existing methods.Comment: submitted to TON (conference version published at IEEE GLOBECOM 2015

Linear Network Coding, Linear Index Coding and Representable Discrete Polymatroids

Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding and representable discrete polymatroids. We consider vector linear solutions of networks over a field $\mathbb{F}_q,$ with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. We define a \textit{discrete polymatroidal} network and show that a linear fractional solution over a field $\mathbb{F}_q,$ exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over $\mathbb{F}_q.$ An algorithm to construct networks starting from certain class of discrete polymatroids is provided. Every representation over $\mathbb{F}_q$ for the discrete polymatroid, results in a linear fractional solution over $\mathbb{F}_q$ for the constructed network. Next, we consider the index coding problem and show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions which are determined by the index coding problem considered. El Rouayheb et. al. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a \textit{perfect linear index coding solution} for an index coding problem obtained from that matroid. We generalize the result of El Rouayheb et. al. by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid.Comment: 24 pages, 6 figures, 4 tables, some sections reorganized, Section VI newly added, accepted for publication in IEEE Transactions on Information Theor

A Linear Network Code Construction for General Integer Connections Based on the Constraint Satisfaction Problem

The problem of finding network codes for general connections is inherently difficult. Resource minimization for general connections with network coding is further complicated. The existing solutions mainly rely on very restricted classes of network codes, and are almost all centralized. In this paper, we introduce linear network mixing coefficients for code constructions of general connections that generalize random linear network coding (RLNC) for multicast connections. For such code constructions, we pose the problem of cost minimization for the subgraph involved in the coding solution and relate this minimization to a Constraint Satisfaction Problem (CSP) which we show can be simplified to have a moderate number of constraints. While CSPs are NP-complete in general, we present a probabilistic distributed algorithm with almost sure convergence in finite time by applying Communication Free Learning (CFL). Our approach allows fairly general coding across flows, guarantees no greater cost than routing, and shows a possible distributed implementation