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An Optimal Sequence Reconstruction Algorithm for Reed-Solomon Codes

Shubhransh Singhvi, Roni Con, Han Mao Kiah, Eitan Yaakobi

TL;DR

For the ubiquitous Reed-Solomon codes, the Koetter-Vardy soft-decoding algorithm is adapted, presenting a reconstruction algorithm capable of correcting beyond Johnson radius.

Abstract

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a scenario where the sender transmits a codeword from some codebook, and the receiver obtains $N$ noisy outputs of the codeword. We study the problem of efficient reconstruction using $N$ outputs that are each corrupted by at most $t$ substitutions. Specifically, for the ubiquitous Reed-Solomon codes, we adapt the Koetter-Vardy soft-decoding algorithm, presenting a reconstruction algorithm capable of correcting beyond Johnson radius. Furthermore, the algorithm uses $\mathcal{O}(nN)$ field operations, where $n$ is the codeword length.

An Optimal Sequence Reconstruction Algorithm for Reed-Solomon Codes

TL;DR

For the ubiquitous Reed-Solomon codes, the Koetter-Vardy soft-decoding algorithm is adapted, presenting a reconstruction algorithm capable of correcting beyond Johnson radius.

Abstract

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a scenario where the sender transmits a codeword from some codebook, and the receiver obtains noisy outputs of the codeword. We study the problem of efficient reconstruction using outputs that are each corrupted by at most substitutions. Specifically, for the ubiquitous Reed-Solomon codes, we adapt the Koetter-Vardy soft-decoding algorithm, presenting a reconstruction algorithm capable of correcting beyond Johnson radius. Furthermore, the algorithm uses field operations, where is the codeword length.
Paper Structure (11 sections, 12 theorems, 26 equations, 1 figure, 3 algorithms)

This paper contains 11 sections, 12 theorems, 26 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Existing Reconstruction Decoders
  3. Decoder based on majority-logic-with-threshold
  4. Brute-force decoder
  5. List-decoding using a single read
  6. Our Decoder
  7. Soft Decoding á la Koetter and Vardy
  8. Constructing the Multiplicity Matrix in Our Settings
  9. Constructing the Multiplicity Matrix Using only Two Reads
  10. Connecting the Distance between the Two Reads with the Decoding Radius
  11. Back to Levenshtein's Reconstruction Problem

Key Result

Lemma 1

Let $e\triangleq \left\lfloor \frac{d-1}{2} \right\rfloor$ and $t \triangleq e + \ell$. Then

Figures (1)

  • Figure 1: Tradeoff between rate $R$ and the fraction of errors that can be corrected. The algorithm that achieves the tradeoff corresponding to \ref{['eq:dec-radius-linear-asym']} has complexity ${\cal O}(nN)$, while the algorithm achieving \ref{['eq:dec-radius-quadratic-asym']} has complexity ${\cal O}(nN^2)$.

Theorems & Definitions (29)

  • Lemma 1: L01A
  • Definition 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Remark 2
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 3: KV03
  • ...and 19 more