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

2026-04-01

AI Summary

This work addresses the long-standing absence of a unified definition and theoretical framework for minimum distance in error correction and detection within nonlinear network coding. By constructing a generalized network channel and coding model, the paper proposes, for the first time, a unified distance theory tailored to joint error correction and detection. It rigorously defines and distinguishes two types of distances, clarifies their interrelationship, and introduces a novel distance metric along with its refined variants. Leveraging information-theoretic and algebraic methods, the study derives fundamental distance bounds and establishes necessary and sufficient conditions for joint error correction and detection capabilities, thereby fully characterizing their interplay and filling a critical gap in the error control theory of nonlinear network coding.

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.

Problem

• generalized network code
• minimum distance
• error correction
• error detection
• nonlinear network codes

Innovation

• generalized network code
• minimum distance
• error correction
• error detection
• nonlinear network coding