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Unique Decoding of Hyperderivative Reed-Solomon Codes

Haojie Gu, Jun Zhang

TL;DR

The paper addresses decoding Hyperderivative Reed-Solomon (HRS) codes under the NRT metric. It extends the Welch-Berlekamp decoding framework to the NRT setting by employing an $E(X)/N(X)$-style relation together with Hermite interpolation, enabling unique decoding up to $e\le\frac{rs-t}{2}$ errors in time $O(r^3s^3)$. The authors show that HRS codes are MDS under the NRT metric with $d_N=rs-t+1$ and provide a concrete algorithm that recovers the transmitted polynomial $P(X)$ from a noisy reception. This work advances efficient decoding in non-Hamming metrics and sets the stage for potential list-decoding extensions and broader applicability of NRT-coded systems.

Abstract

Error-correcting codes are combinatorial objects designed to cope with the problem of reliable transmission of information on a noisy channel. A fundamental problem in coding theory and practice is to efficiently decode the received word with errors to obtain the transmitted codeword. In this paper, we consider the decoding problem of Hyperderivative Reed-Solomon (HRS) codes with respect to the NRT metric. Specifically, we propose a Welch-Berlekamp algorithm for the unique decoding of NRT HRS codes.

Unique Decoding of Hyperderivative Reed-Solomon Codes

TL;DR

The paper addresses decoding Hyperderivative Reed-Solomon (HRS) codes under the NRT metric. It extends the Welch-Berlekamp decoding framework to the NRT setting by employing an -style relation together with Hermite interpolation, enabling unique decoding up to errors in time . The authors show that HRS codes are MDS under the NRT metric with and provide a concrete algorithm that recovers the transmitted polynomial from a noisy reception. This work advances efficient decoding in non-Hamming metrics and sets the stage for potential list-decoding extensions and broader applicability of NRT-coded systems.

Abstract

Error-correcting codes are combinatorial objects designed to cope with the problem of reliable transmission of information on a noisy channel. A fundamental problem in coding theory and practice is to efficiently decode the received word with errors to obtain the transmitted codeword. In this paper, we consider the decoding problem of Hyperderivative Reed-Solomon (HRS) codes with respect to the NRT metric. Specifically, we propose a Welch-Berlekamp algorithm for the unique decoding of NRT HRS codes.
Paper Structure (7 sections, 7 theorems, 42 equations, 1 algorithm)

This paper contains 7 sections, 7 theorems, 42 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Hyperderivatives
  4. The NRT Metric
  5. Hyperderivative Reed-Solomon Codes
  6. The Welch-Berlekamp Algorithm of NRT HRS codes
  7. Conclusion

Key Result

Lemma 2.2

For any $A(X),B(X)\in \mathbb{F}_{q}[X]$ and $\ell\geq 0$, we have

Theorems & Definitions (13)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Lemma 2.2: dvir2013extensions
  • Theorem 2.3: von2003modern
  • Lemma 2.4: skriganov2001coding
  • Theorem 2.5: skriganov2001coding
  • Example 3.1
  • Lemma 3.2
  • proof
  • ...and 3 more