Bounds on Unique-Neighbor Codes
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Let A be an m× n parity check matrix of a binary code of length n, rate R, and distance δ n. This means that for every δ n >k>0, every m× k submatrix of A has a row of odd weight. Message-passing decoding algorithms require the stronger unique neighbor property. Namely, that every such submatrix has a row of weight 1. It is well known that if δ≥1/2, then R=o_n(1) whereas for every 1/2>δ there exist linear codes of length n→∞ and distance > δ n with R bounded away from zero. We believe that the unique neighbor property entails sharper upper bounds on the rate. Concretely, we conjecture that for a proper choice of f(n) = o(n) and some constant ϵ_0 > 0, every n×(n+f(n)) binary matrix has an n× k submatrix with (1/2 -ϵ_0)n≥ k >0 where no row weighs 1. We make several contributions to the study of this conjecture.
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