On the Feasibility of Network Alignment for Three-Source Three-Destination Multiple Unicast Networks with Delays

TL;DR

This paper demonstrates that the necessary and sufficient conditions for the feasibility of Precoding Based Network Alignment (PBNA) in three-source three-destination multiple unicast networks with delays are established using a transform approach and block time varying local encoding kernels.

Contribution

It extends previous work by proving the necessary and sufficient conditions for PBNA feasibility in delayed networks using the transform approach.

Findings

Conditions of Meng et al. are necessary and sufficient for 3-S 3-D MUN-D.

Transform approach effectively analyzes delayed network coding.

PBNA feasibility characterized by finite, verifiable conditions.

Abstract

A transform approach to network coding was introduced by Bavirisetti et al. (arXiv:1103.3882v3 [cs.IT]) as a tool to view wireline networks with delays as $k$-instantaneous networks (for some large $k$). When the local encoding kernels (LEKs) of the network are varied with every time block of length $k > 1$, the network is said to use block time varying LEKs. In this work, we propose a Precoding Based Network Alignment (PBNA) scheme based on transform approach and block time varying LEKs for three-source three-destination multiple unicast network with delays (3-S 3-D MUN-D). In a recent work, Meng et al. (arXiv:1202.3405v1 [cs.IT]) reduced the infinite set of sufficient conditions for feasibility of PBNA in a three-source three-destination instantaneous multiple unicast network as given by Das et al. (arXiv:1008.0235v1 [cs.IT]) to a finite set and also showed that the conditions are necessary. We show that the conditions of Meng et al. are also necessary and sufficient conditions for feasibility of PBNA based on transform approach and block time varying LEKs for 3-S 3-D MUN-D.