On Separation, Randomness and Linearity for Network Codes over Finite Fields

TL;DR

This paper investigates the conditions under which source-channel separation holds in network coding over finite fields, demonstrating the optimality of linear codes in various network scenarios and discussing the implications for network decomposability.

Contribution

It establishes the optimality of linear codes for multiple network channels and highlights the limitations of network decomposition for coding strategies.

Findings

Separation holds for certain canonical networks over finite fields.

Linear codes are sufficient for optimal coding in multiple network scenarios.

Decomposing networks into subnetworks may not always be effective.

Abstract

We examine the issue of separation and code design for networks that operate over finite fields. We demonstrate that source-channel (or source-network) separation holds for several canonical network examples like the noisy multiple access channel and the erasure degraded broadcast channel, when the whole network operates over a common finite field. This robustness of separation is predicated on the fact that noise and inputs are independent, and we examine the failure of separation when noise is dependent on inputs in multiple access channels. Our approach is based on the sufficiency of linear codes. Using a simple and unifying framework, we not only re-establish with economy the optimality of linear codes for single-transmitter, single-receiver channels and for Slepian-Wolf source coding, but also establish the optimality of linear codes for multiple access and for erasure degraded broadcast channels. The linearity allows us to obtain simple optimal code constructions and to study capacity regions of the noisy multiple access and the degraded broadcast channel. The linearity of both source and network coding blurs the delineation between source and network codes. While our results point to the fact that separation of source coding and channel coding is optimal in some canonical networks, we show that decomposing networks into canonical subnetworks may not be effective. Thus, we argue that it may be the lack of decomposability of a network into canonical network modules, rather than the lack of separation between source and channel coding, that presents major challenges for coding over networks.