On Minimum Distances for Error Correction and Detection of Generalized Network Code

TL;DR

This paper introduces a generalized framework for network codes, defining and analyzing distances for error correction and detection, especially in nonlinear codes, and establishing bounds and relationships among these distances.

Contribution

It systematically defines and characterizes error correction and detection distances in generalized network codes, extending understanding beyond linear codes.

Findings

Established a scheme for generalized network channels and codes.

Defined and refined distances for error correction and detection.

Proved bounds relating various error correction and detection distances.

Abstract

It is well known that the minimum distance for linear network codes plays the same role as the minimum distance for classical error control codes. However, Yang and Yeung (2008) discovered that for nonlinear network codes, the minimum distance for error correction is not always the same as the minimum distance for error detection. Inspired by the idea that the channel will affect the distances between the codewords, we establish the scheme of a generalized network channel and a generalized network code. Then, we systematically define the distances for error correction and error detection under the scheme of the generalized network code. We consider the joint error correction and detection in the generalized network code and obtain a complete characterization by introducing a distance and its refined version for this purpose. We enhance our understanding of the relation between various distances for error correction and detection in generalized network codes by proving some bounds on these distances.