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Trace Reconstruction of First-Order Reed-Muller Codewords Using Run Statistics

Shiv Pratap Singh Rathore, Navin Kashyap

TL;DR

We address exact reconstruction of a binary sequence from multiple traces through an i.i.d. deletion channel, focusing on codewords of the binary first-order Reed–Muller code RM$(m,1)$. The authors derive an explicit formula for the expected total number of runs $\mathbb{E}[R_x]$ in a trace as a function of the codeword $x$ and the deletion rate $q$, and specialize to $q=\tfrac{1}{2}$ to enable reconstruction. They prove that, w.h.p., a codeword from RM$(m,1)$ can be recovered from $\tilde{O}(n^2)$ traces, using a two-phase strategy: infer the first bit from $O(\log n)$ traces and then separate codewords with the same first bit by comparing the empirical mean run count against a guaranteed gap $\Delta\ge 0.028$. This work demonstrates a purely statistical use of run counts for trace reconstruction and clarifies the limitations for higher-order RM codes, outlining avenues for refining the statistic or combining it with other features.

Abstract

In this paper, we derive an expression for the expected number of runs in a trace of a binary sequence $x \in \{0,1\}^n$ obtained by passing $x$ through a deletion channel that independently deletes each bit with probability $q$. We use this expression to show that if $x$ is a codeword of a first-order Reed-Muller code, and the deletion probability $q$ is 1/2, then $x$ can be reconstructed, with high probability, from $\tilde{O}(n^2)$ many of its traces.

Trace Reconstruction of First-Order Reed-Muller Codewords Using Run Statistics

TL;DR

We address exact reconstruction of a binary sequence from multiple traces through an i.i.d. deletion channel, focusing on codewords of the binary first-order Reed–Muller code RM. The authors derive an explicit formula for the expected total number of runs in a trace as a function of the codeword and the deletion rate , and specialize to to enable reconstruction. They prove that, w.h.p., a codeword from RM can be recovered from traces, using a two-phase strategy: infer the first bit from traces and then separate codewords with the same first bit by comparing the empirical mean run count against a guaranteed gap . This work demonstrates a purely statistical use of run counts for trace reconstruction and clarifies the limitations for higher-order RM codes, outlining avenues for refining the statistic or combining it with other features.

Abstract

In this paper, we derive an expression for the expected number of runs in a trace of a binary sequence obtained by passing through a deletion channel that independently deletes each bit with probability . We use this expression to show that if is a codeword of a first-order Reed-Muller code, and the deletion probability is 1/2, then can be reconstructed, with high probability, from many of its traces.
Paper Structure (16 sections, 5 theorems, 39 equations, 1 table)

This paper contains 16 sections, 5 theorems, 39 equations, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Binary First-Order Reed-Muller Codes
  4. Chernoff Bounds
  5. Hoeffding's Inequality
  6. Expected Number of Runs in a Trace
  7. Trace Reconstruction for $\mathrm{RM}(m,1)$
  8. Proof of Theorem \ref{['thm:RMm1']}
  9. Proof of Proposition 1
  10. Proof of Lemma \ref{['lem:conditions']}
  11. Conclusion
  12. Proof of Lemma \ref{['lemma_update']}(b)
  13. Proofs of (C1)--(C3) in Cases 2--4
  14. Proof of (C1)--(C3) in Case 2
  15. Proof of (C1)--(C3) in Case 3
  16. ...and 1 more sections

Key Result

Theorem 1

Given a binary sequence $x$, the expected numbers of runs of $\mathsf{0}$s and $\mathsf{1}$s in a trace $T_x$ obtained by independently deleting each bit in $x$ with probability $q$ are given by and respectively. Consequently, $\mathbb{E}[R_x] = \mathbb{E}[R_{x,\mathsf{0}}]+\mathbb{E}[R_{x,\mathsf{1}}]$ is given by

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Theorem 2
  • Proposition 1
  • Claim 1
  • proof
  • Claim 2
  • proof
  • Lemma 3
  • Lemma 4
  • ...and 2 more