We study the density of the weights of Generalized Reed--Muller codes. Let @math denote the code of multivariate polynomials over @math in @math variables of total degree at most @math . We consider the case of fixed degree @math , when we let the number of variables @math tend to infinity. We prove that the set of relative weights of codewords is quite sparse: for every @math which is not rational of the form @math , there exists an interval around @math in which no relative weight exists, for any value of @math . This line of research is to the best of our knowledge new, and complements the traditional lines of research, which focus on the weight distribution and the divisibility properties of the weights. Equivalently, we study distributions taking values in a finite field, which can be approximated by distributions coming from constant degree polynomials, where we do not bound the number of variables. We give a complete characterization of all such distributions.
The weight distribution of @math is the number of codewords below a certain weight. The case of @math , i.e. of linear functions, is trivial, since all non-constant codewords have the same weight. The case of @math , i.e. of quadratic functions, is also fully understood. A theorem of Dixon @cite_2 gives a canonical characterization of quadratic functions, and in particular gives the possible weights and the weight distribution of quadratic functions. By the McWilliams identity, this characterize the weight distribution of their dual codes, which are @math and @math . These are, to the best of our knowledge, the only (non-trivial) orders for which complete characterization the weights of Generalized Reed--Muller codes is known. For other orders, complete characterization is known only for specific values of @math . For example, for cubics the record is the work of Sugita, Kasami and Fujiwara @cite_13 , characterizing the weight distribution for @math .
@cite_13 @cite_2
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This article offers a simplified approach to the distribution theory of randomly weighted averages or @math -means @math , for a sequence of i.i.d.random variables @math , and independent random weights @math with @math and @math . The collection of distributions of @math , indexed by distributions of @math , is shown to encode Kingman's partition structure derived from @math . For instance, if @math has Bernoulli @math distribution on @math , the @math th moment of @math is a polynomial function of @math which equals the probability generating function of the number @math of distinct values in a sample of size @math from @math : @math . This elementary identity illustrates a general moment formula for @math -means in terms of the partition structure associated with random samples from @math , first developed by Diaconis and Kemperman (1996) and Kerov (1998) in terms of random permutations. As shown by Tsilevich (1997) if the partition probabilities factorize in a way characteristic of the generalized Ewens sampling formula with two parameters @math , found by Pitman (1992), then the moment formula yields the Cauchy-Stieltjes transform of an @math mean. The analysis of these random means includes the characterization of @math -means, known as Dirichlet means, due to Von Neumann (1941), Watson (1956) and Cifarelli and Regazzini (1990) and generalizations of L 'evy's arcsine law for the time spent positive by a Brownian motion, due to Darling (1949) Lamperti (1958) and Barlow, Pitman and Yor (1989). A @math -modal probability distribution over the discrete domain @math is one whose histogram has at most @math "peaks" and "valleys." Such distributions are natural generalizations of monotone ( @math ) and unimodal ( @math ) probability distributions, which have been intensively studied in probability theory and statistics. In this paper we consider the problem of (i.e., performing density estimation of) an unknown @math -modal distribution with respect to the @math distance. The learning algorithm is given access to independent samples drawn from an unknown @math -modal distribution @math , and it must output a hypothesis distribution @math such that with high probability the total variation distance between @math and @math is at most @math Our main goal is to obtain algorithms for this problem that use (close to) an information-theoretically optimal number of samples. We give an efficient algorithm for this problem that runs in time @math . For @math , the number of samples used by our algorithm is very close (within an @math factor) to being information-theoretically optimal. Prior to this work computationally efficient algorithms were known only for the cases @math Birge:87b,Birge:97 . A novel feature of our approach is that our learning algorithm crucially uses a new algorithm for as a key subroutine. The learning algorithm uses the property tester to efficiently decompose the @math -modal distribution into @math (near-)monotone distributions, which are easier to learn. We present a simple distributed @math -approximation algorithm for maximum weight independent set (MaxIS) in the @math model which completes in @math rounds, where @math is the maximum degree, @math is the number of rounds needed to compute a maximal independent set (MIS) on @math , and @math is the maximum weight of a node. Whether our algorithm is randomized or deterministic depends on the MIS algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in @math rounds, where @math is the number of nodes. We also present a deterministic @math -round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a @math -approximation for maximum weight matching without incurring any additional round penalty in the @math model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of for which we describe a mechanism that allows the simulation to run in the @math model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ( @math ) allows us to devise a distributed algorithm requiring @math rounds for any constant @math . For the unweighted case, we can even obtain a @math -approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on @math . We investigate the approximability of several classes of real-valued functions by functions of a small number of variables ( juntas ). Our main results are tight bounds on the number of variables required to approximate a function @math within @math -error @math over the uniform distribution: 1. If @math is submodular, then it is @math -close to a function of @math variables. This is an exponential improvement over previously known results. We note that @math variables are necessary even for linear functions. 2. If @math is fractionally subadditive (XOS) it is @math -close to a function of @math variables. This result holds for all functions with low total @math -influence and is a real-valued analogue of Friedgut's theorem for boolean functions. We show that @math variables are necessary even for XOS functions. As applications of these results, we provide learning algorithms over the uniform distribution. For XOS functions, we give a PAC learning algorithm that runs in time @math . For submodular functions we give an algorithm in the more demanding PMAC learning model (Balcan and Harvey, 2011) which requires a multiplicative @math factor approximation with probability at least @math over the target distribution. Our uniform distribution algorithm runs in time @math . This is the first algorithm in the PMAC model that over the uniform distribution can achieve a constant approximation factor arbitrarily close to 1 for all submodular functions. As follows from the lower bounds in (, 2013) both of these algorithms are close to optimal. We also give applications for proper learning, testing and agnostic learning with value queries of these classes.