Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes
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We study the following natural question on random sets of points in 𝔽_2^m: Given a random set of k points Z={z_1, z_2, …, z_k}⊆𝔽_2^m, what is the dimension of the space of degree at most r multilinear polynomials that vanish on all points in Z? We show that, for r ≤γ m (where γ > 0 is a small, absolute constant) and k = (1-ϵ) ·m≤ r for any constant ϵ > 0, the space of degree at most r multilinear polynomials vanishing on a random set Z = {z_1,…, z_k} has dimension exactly m≤ r - k with probability 1 - o(1). This bound shows that random sets have a much smaller space of degree at most r multilinear polynomials vanishing on them, compared to the worst-case bound (due to Wei (IEEE Trans. Inform. Theory, 1991)) of m≤ r - log_2 k≤ r≫m≤ r - k. Using this bound, we show that high-degree Reed-Muller codes (RM(m,d) with d > (1-γ) m) "achieve capacity" under the Binary Erasure Channel in the sense that, for any ϵ > 0, we can recover from (1 - ϵ) ·m≤ m-d-1 random erasures with probability 1 - o(1). This also implies that RM(m,d) is also efficiently decodable from ≈m≤ m-(d/2) random errors for the same range of parameters.
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