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List Decoding Reed--Solomon Codes in the Lee, Euclidean, and Other Metrics

Chris Peikert, Alexandra Veliche Hostetler

TL;DR

This work extends list decoding of Generalized Reed–Solomon codes to $\ell_p$ metrics with $0< p \le 2$, notably the Lee ($p=1$) and Euclidean ($p=2$) cases, by embedding received words into smooth weight vectors and applying the Guruswami–Sudan soft-decision decoder. The core method combines Fourier-analytic lattice techniques with a flexible smoothing function $f^{(p)}$ to bound codeword correlations and optimize decoding parameters, yielding improved distance–rate tradeoffs over prior $\ell_1$/$\ell_2$ decoders and enabling decoding at arbitrarily large distances for small rates. The paper also derives lower bounds on the $\ell_1$ and $\ell_2$ minimum distances for a natural subclass of GRS codes, yielding conditions under which the list-decoder effectively acts as a unique decoder. In addition to worst-case guarantees, average-case results under Laplacian and Gaussian errors show that the algorithm can support even larger rates in probabilistic channels. Overall, the approach broadens RS code decoding to more realistic error models, providing stronger theoretical guarantees and practical decoding performance in channels with structured noise.

Abstract

Reed--Solomon error-correcting codes are ubiquitous across computer science and information theory, with applications in cryptography, computational complexity, communication and storage systems, and more. Most works on efficient error correction for these codes, like the celebrated Berlekamp--Welch unique decoder and the (Guruswami--)Sudan list decoders, are focused on measuring error in the Hamming metric, which simply counts the number of corrupted codeword symbols. However, for some applications, other metrics that depend on the specific values of the errors may be more appropriate. This work gives a polynomial-time algorithm that list decodes (generalized) Reed--Solomon codes over prime fields in $\ell_p$ (semi)metrics, for any $0 < p \leq 2$. Compared to prior algorithms for the Lee ($\ell_1$) and Euclidean ($\ell_2$) metrics, ours decodes to arbitrarily large distances (for correspondingly small rates), and has better distance-rate tradeoffs for all decoding distances above some moderate thresholds. We also prove lower bounds on the $\ell_{1}$ and $\ell_{2}$ minimum distances of a certain natural subclass of GRS codes, which establishes that our list decoder is actually a unique decoder for many parameters of interest. Finally, we analyze our algorithm's performance under random Laplacian and Gaussian errors, and show that it supports even larger rates than for corresponding amounts of worst-case error in $\ell_{1}$ and $\ell_{2}$ (respectively).

List Decoding Reed--Solomon Codes in the Lee, Euclidean, and Other Metrics

TL;DR

This work extends list decoding of Generalized Reed–Solomon codes to metrics with , notably the Lee () and Euclidean () cases, by embedding received words into smooth weight vectors and applying the Guruswami–Sudan soft-decision decoder. The core method combines Fourier-analytic lattice techniques with a flexible smoothing function to bound codeword correlations and optimize decoding parameters, yielding improved distance–rate tradeoffs over prior / decoders and enabling decoding at arbitrarily large distances for small rates. The paper also derives lower bounds on the and minimum distances for a natural subclass of GRS codes, yielding conditions under which the list-decoder effectively acts as a unique decoder. In addition to worst-case guarantees, average-case results under Laplacian and Gaussian errors show that the algorithm can support even larger rates in probabilistic channels. Overall, the approach broadens RS code decoding to more realistic error models, providing stronger theoretical guarantees and practical decoding performance in channels with structured noise.

Abstract

Reed--Solomon error-correcting codes are ubiquitous across computer science and information theory, with applications in cryptography, computational complexity, communication and storage systems, and more. Most works on efficient error correction for these codes, like the celebrated Berlekamp--Welch unique decoder and the (Guruswami--)Sudan list decoders, are focused on measuring error in the Hamming metric, which simply counts the number of corrupted codeword symbols. However, for some applications, other metrics that depend on the specific values of the errors may be more appropriate. This work gives a polynomial-time algorithm that list decodes (generalized) Reed--Solomon codes over prime fields in (semi)metrics, for any . Compared to prior algorithms for the Lee () and Euclidean () metrics, ours decodes to arbitrarily large distances (for correspondingly small rates), and has better distance-rate tradeoffs for all decoding distances above some moderate thresholds. We also prove lower bounds on the and minimum distances of a certain natural subclass of GRS codes, which establishes that our list decoder is actually a unique decoder for many parameters of interest. Finally, we analyze our algorithm's performance under random Laplacian and Gaussian errors, and show that it supports even larger rates than for corresponding amounts of worst-case error in and (respectively).

Paper Structure

This paper contains 28 sections, 28 theorems, 94 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Technical Overview
  4. Preliminaries
  5. Linear Codes
  6. Lattices
  7. Fourier Analysis
  8. Lattice Roughness
  9. List-Decoding Reed--Solomon Codes
  10. Soft-Decision Decoding
  11. From Received Words to Weight Vectors
  12. Main Theorem
  13. Average-Case Decoding
  14. General $\ell_{p}$ (Semi)Metrics
  15. Worst-Case Decoding
  16. ...and 13 more sections

Key Result

lemma 1

Let $X_{1}, \ldots, X_{n}$ be independent identically distributed random variables in $[0,1]$ with common expectation $\mu = \mathop{\mathrm{\mathbb{E}}}\limits[X_{i}]$. Then for any $\gamma \geq 0$,

Figures (1)

  • Figure 1: Plots of the adjusted rate $R^{*,(p)}$, as a function of the $\ell_{p}$ relative decoding distance $\delta = d/n^{1/p}$ or corresponding channel error width $r = p^{1/p} \cdot c_{p} \cdot \delta$, for which our algorithm can list decode prime-field GRS codes in the worst case (wc) or average case (ac), respectively, for $p=2$ (left) and $p=1$ (right). (For simplicity, these plots assume a field size $q \gg \delta, r$.) For comparison, also shown are the corresponding functions from the prior work on decoding GRS codes in these metrics: DBLP:journals/tit/MookP22 is for list decoding in $\ell_{2}$, and DBLP:journals/tit/RothS94wu03:_lee_bch_reed_solomon are respectively for unique and list decoding in the $\ell_{1}$ (Lee) metric, but only for discrete (integer) error. Also shown are rate bounds $R^{(p)}_{\text{uniq}}$ for which decoding to $\ell_{p}$ relative distance $\delta$ is guaranteed to yield a unique codeword, for a certain natural subclass of GRS codes. (See lem:ell2-min-dist,lem:ell1-min-dist and the discussions thereafter.)

Theorems & Definitions (62)

  • lemma 1: Hoeffding's Inequality
  • definition 1: (Generalized) Reed--Solomon code
  • definition 2: Lattice, Basis
  • definition 3: Determinant
  • definition 4: Dual lattice
  • lemma 2
  • proof
  • lemma 3: Multiplicativity
  • lemma 4: Time-scaling property
  • lemma 5: Time-shift property
  • ...and 52 more