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

Yulin Chen, Raymond W. Yeung

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.

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

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.

Paper Structure

This paper contains 17 sections, 25 theorems, 114 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Generalized Network Channel and Code
  3. Classical Block Codes, Coherent Network Coding, and Noncoherent Network Coding
  4. Block Code over the Classical Point-to-Point Channel
  5. Network Code over a Coherent Network Channel
  6. Rank Metric Code over a Non-coherent Linear Network Channel
  7. Sum-rank Metric Code over a Non-coherent linear Network Channel
  8. Error Correction And Detection Distances
  9. Two Distances for Error Correction And Detection
  10. Relations between $d_0^{\min}$ and $d_1^{\min}$
  11. Distance for Joint Error Correction and Detection
  12. Relations Between $d_0^{\min}$, $d_1^{\min}$ and $d_2^{\min}$
  13. Refined Distance For Joint Error Correction
  14. A Toy Example
  15. Conclusion
  16. ...and 2 more sections

Key Result

Theorem 4.5

Let $({\cal C}, F)$ be a generalized network code. If $F$ is error-linear, then $D_0(x_1,x_2)=D_1(x_1,x_2)$ for all $x_1,x_2\in\mathcal{C}$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: example network from WeightProperty. $s$ is the source node and $t$ is the sink node.

Theorems & Definitions (34)

  • Definition 4.1: Metric
  • Definition 4.2: Decoding BallWeightProperty
  • Definition 4.3: Error Correction DistanceWeightProperty
  • Definition 4.4: Error Detection DistanceWeightProperty
  • Theorem 4.5
  • Remark 4.6
  • Corollary 4.7
  • Theorem 4.8
  • Corollary 4.9
  • Definition 5.1: Joint error correction and detection
  • ...and 24 more