Linear Network Error Correction Multicast/Broadcast/Dispersion Codes

TL;DR

This paper generalizes linear multicast, broadcast, and dispersion network codes to include error correction, establishes bounds for their optimality, and proves the existence of maximum distance separable codes with constructive methods.

Contribution

It introduces a unified framework for linear network error correction codes across multicast, broadcast, and dispersion scenarios, including bounds and existence proofs.

Findings

Established Singleton bounds for new code classes

Proved existence of optimal MDS codes

Proposed a constructive algorithm for code design

Abstract

In this paper, for the purposes of information transmission and network error correction simultaneously, three classes of important linear network codes in network coding, linear multicast/broadcast/dispersion codes are generalized to linear network error correction coding, i.e., linear network error correction multicast/broadcast/dispersion codes. We further propose the (weakly, strongly) extended Singleton bounds for these new classes of codes, and define the optimal codes satisfying the corresponding Singleton bounds with equality, which are called multicast/broadcast/dispersion MDS codes respectively. The existence of such codes are proved by an algebraic method and one kind of constructive algorithm is also proposed.