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

Shiv Pratap Singh Rathore; Navin Kashyap

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arXiv · 2501.11393

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

Shiv Pratap Singh Rathore·Navin Kashyap

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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.

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Shiv Pratap Singh Rathore, Navin Kashyap. 2025-01-20. Trace Reconstruction of First-Order Reed-Muller Codewords Using Run Statistics. https://arxiv.org/abs/2501.11393

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