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

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.