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

TL;DR

This paper develops a method to reconstruct first-order Reed-Muller codewords from multiple traces affected by deletions, using run statistics, and demonstrates high-probability reconstruction with quadratic trace complexity.

Contribution

It introduces a novel analysis of run statistics in traces and shows efficient reconstruction of Reed-Muller codewords under deletion channels.

Findings

Reconstruction possible with high probability at O(n^2) traces

Derived an expression for expected runs in traces

Applicable to deletion probability of 1/2

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.