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Bounds and Codes for General Phased Burst Errors

Sebastian Bitzer, Andrea Di Giusto, Alberto Ravagnani, Eitan Yaakobi

TL;DR

By modeling PBEs as an error set in an adversarial channel, bounds on the maximal size of codes that can correct them are investigated, and asymptotically good PBE-correcting codes are constructed, recovering a classical construction in a specific problem instance.

Abstract

Phased burst errors (PBEs) are bursts of errors occurring at one or more known locations. The correction of PBEs is a classical topic in coding theory, with prominent applications such as the design of array codes for memory systems or distributed storage. We propose a general yet fine-grained approach to this problem, accounting not only for the number of bursts but also the error structure in each burst. By modeling PBEs as an error set in an adversarial channel, we investigate bounds on the maximal size of codes that can correct them. The PBE-correction capability of generalized concatenated codes is analyzed, and asymptotically good PBE-correcting codes are constructed, recovering a classical construction in a specific problem instance.

Bounds and Codes for General Phased Burst Errors

TL;DR

By modeling PBEs as an error set in an adversarial channel, bounds on the maximal size of codes that can correct them are investigated, and asymptotically good PBE-correcting codes are constructed, recovering a classical construction in a specific problem instance.

Abstract

Phased burst errors (PBEs) are bursts of errors occurring at one or more known locations. The correction of PBEs is a classical topic in coding theory, with prominent applications such as the design of array codes for memory systems or distributed storage. We propose a general yet fine-grained approach to this problem, accounting not only for the number of bursts but also the error structure in each burst. By modeling PBEs as an error set in an adversarial channel, we investigate bounds on the maximal size of codes that can correct them. The PBE-correction capability of generalized concatenated codes is analyzed, and asymptotically good PBE-correcting codes are constructed, recovering a classical construction in a specific problem instance.
Paper Structure (14 sections, 9 theorems, 35 equations, 2 figures)

This paper contains 14 sections, 9 theorems, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The PBE Adversarial Channel
  3. Notation
  4. Problem Statement
  5. Bounds on the Maximal Code Size
  6. Code Construction
  7. Proofs Related to the Proposed Bounds
  8. More admissible sequences of error sets and channels
  9. Proof of \ref{['corr:HPBE_H', 'corr:HPBE_GV']}
  10. Proof of \ref{['comparison:PBE_vs_Hamm']}
  11. Proofs Related to the Code Construction
  12. Properties of Construction \ref{['const:2lvl']}
  13. Properties of Construction \ref{['const:3lvl']}
  14. Proof of \ref{['comparison:2lvl_3lvl_gv']}

Key Result

Lemma 3

For any channel $\Omega$ over $\mathcal{V}$,

Figures (2)

  • Figure 1: Two possible Hamming PBEs with $n = 5$, $t=2$, $m=6$, and $w=2$. That is, $\mathcal{E}_1 = \{\mathbf0\}\subset\mathcal{E}_2=B_{2}(5,2)$.
  • Figure 2: Comparison of the rates achieved by \ref{['const:2lvl']} and \ref{['const:3lvl']} with the GV and Hamming bound for $q=2$.

Theorems & Definitions (27)

  • Definition 1
  • Example 2
  • Lemma 3: Proposition 1, loeliger1994basic
  • Definition 4
  • Theorem 5: Theorem 1, loeliger1994basic
  • Example 6
  • Definition 7
  • Definition 8
  • Definition 9: Hamming PBC/PBEs
  • Definition 10: Admissible sequences of PBE sets/channels
  • ...and 17 more