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We study the problem of how well a typical multivariate polynomial can be approximated by lower-degree polynomials over $${\mathbb F}$$ . We prove that almost all degree d polynomials have only an exponentially small correlation with all polynomials of degree at most d − 1, for all degrees d up to Θ( n). That is, a random degree d polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low-degree polynomial. Recently, several results regarding the weight distribution of Reed-Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed-Muller codes. [ABSTRACT FROM AUTHOR]

Moran, Naor, and Segev have asked what is the minimal r = r(n, k) for which there exists an (n, k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2,. . . , n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call α-fraction k-multiuser tracing ((k, α)-FUT (fraction user-tracing) families). We show that r(n, k) = ϴ(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, α)-FUT families and by constructing an (n, k) -monotone encoding of length O( k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n + O(1), which is optimal up to an additive constant. [ABSTRACT FROM AUTHOR]