Refined Coding Bounds and Code Constructions for Coherent Network Error Correction

TL;DR

This paper refines classical coding bounds for coherent network error correction, providing tight bounds and construction algorithms that enhance error correction capabilities based on network flow and coding strategies.

Contribution

It introduces refined bounds for coherent network error correction and presents two algorithms to construct codes achieving these bounds, linking network flow with error correction.

Findings

Refined Hamming, Singleton, and Gilbert-Varshamov bounds for network coding.

Tightness of the refined Singleton bound demonstrated by two construction algorithms.

Higher flow nodes can achieve better error correction capabilities.

Abstract

Coherent network error correction is the error-control problem in network coding with the knowledge of the network codes at the source and sink nodes. With respect to a given set of local encoding kernels defining a linear network code, we obtain refined versions of the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for coherent network error correction. Similar to its classical counterpart, this refined Singleton bound is tight for linear network codes. The tightness of this refined bound is shown by two construction algorithms of linear network codes achieving this bound. These two algorithms illustrate different design methods: one makes use of existing network coding algorithms for error-free transmission and the other makes use of classical error-correcting codes. The implication of the tightness of the refined Singleton bound is that the sink nodes with higher maximum flow values can have higher error correction capabilities.