A Transform Approach to Linear Network Coding for Acyclic Networks with Delay

TL;DR

This paper introduces a transform-based method using finite-field DFT to analyze and design linear network codes for acyclic networks with delays, enabling simplified analysis and improved rate achievement in certain multicast scenarios.

Contribution

It proposes a novel transform approach converting delayed acyclic networks into instantaneous networks, establishing equivalence with traditional methods and enabling new coding strategies.

Findings

Transform method simplifies network coding analysis with delays.

Existence of network codes in transform approach matches traditional approach.

Achieves half the min-cut rate in certain delayed unicast networks.

Abstract

The algebraic formulation for linear network coding in acyclic networks with the links having integer delay is well known. Based on this formulation, for a given set of connections over an arbitrary acyclic network with integer delay assumed for the links, the output symbols at the sink nodes, at any given time instant, is a \mathbb{F}_{q}$-linear combination of the input symbols across different generations, where $\mathbb{F}_{q}$ denotes the field over which the network operates. We use finite-field discrete fourier transform (DFT) to convert the output symbols at the sink nodes, at any given time instant, into a $\mathbb{F}_{q}$-linear combination of the input symbols generated during the same generation. We call this as transforming the acyclic network with delay into {\em $n$-instantaneous networks} ($n$ is sufficiently large). We show that under certain conditions, there exists a network code satisfying sink demands in the usual (non-transform) approach if and only if there exists a network code satisfying sink demands in the transform approach. Furthermore, we show that the transform method (along with the use of alignment strategies) can be employed to achieve half the rate corresponding to the individual source-destination min-cut (which are assumed to be equal to 1) for some classes of three-source three-destination unicast network with delays, when the zero-interference conditions are not satisfied.