A Matroid Framework for Noncoherent Random Network Communications

Abstract

Models for noncoherent error control in random linear network coding (RLNC) and store and forward (SAF) have been recently proposed. In this paper, we model different types of random network communications as the transmission of flats of matroids. This novel framework encompasses RLNC and SAF and allows us to introduce a novel protocol, referred to as random affine network coding (RANC), based on affine combinations of packets. Although the models previously proposed for RLNC and SAF only consider error control, using our framework, we first evaluate and compare the performance of different network protocols in the error-free case. We define and determine the rate, average delay, and throughput of such protocols, and we also investigate the possibilities of partial decoding before the entire message is received. We thus show that RANC outperforms RLNC in terms of data rate and throughput thanks to a more efficient encoding of messages into packets. Second, we model the possible alterations of a message by the network as an operator channel, which generalizes the channels proposed for RLNC and SAF. Error control is thus reduced to a coding-theoretic problem on flats of a matroid, where two distinct metrics can be used for error correction. We study the maximum cardinality of codes on flats in general, and codes for error correction in RANC in particular. We finally design a class of nearly optimal codes for RANC based on rank metric codes for which we propose a low-complexity decoding algorithm. The gain of RANC over RLNC is thus preserved with no additional cost in terms of complexity.

Probability of Partially Decoding Network-Coded Messages

“…Theorem 1 can be seen as a special case of [4,Proposition 6]. Whereas the proof in [4] uses elements of matroid theory, our paper proposes an alternative and more intuitive proof strategy. Corollary 1.…”

Entropy of Closure Operators and Network Coding Solvability

“…We would like to explain the significance of flats in matroids for random network coding. A model for noncoherent random network coding based on matroids is proposed in [21], which generalises routing (a special case for the uniform matroid), linear network coding (the projective geometry) and affine network coding (the affine geometry). In order to combine the messages they receive, the intermediate nodes select a random element from the closure of the received messages.…”

A Survey of Linear Network Coding and Network Error Correction Code Constructions and Algorithms

“…Linear correlation is indeed exploited by random linear codes, as discussed in [31]. A randomized approach based on matroids theory that encompasses multiple sources was presented in [32].…”