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
1797136128 1606480398
Extractors are functions which are able to “extract” random bits from arbitrary distributions which “contain” sufficient randomness. Explicit constructions of extractors have many applications in complexity theory and combinatorics. This manuscript is a survey of recent developments in extractors and focuses on explicit constructions of extractors following Trevisan’s breakthrough result [Tre99]. Linear Codes. Nonlinear Codes, Hadamard Matrices, Designs and the Golay Code. An Introduction to BCH Codes and Finite Fields. Finite Fields. Dual Codes and Their Weight Distribution. Codes, Designs and Perfect Codes. Cyclic Codes. Cyclic Codes: Idempotents and Mattson-Solomon Polynomials. BCH Codes. Reed-Solomon and Justesen Codes. MDS Codes. Alternant, Goppa and Other Generalized BCH Codes. Reed-Muller Codes. First-Order Reed-Muller Codes. Second-Order Reed-Muller, Kerdock and Preparata Codes. Quadratic-Residue Codes. Bounds on the Size of a Code. Methods for Combining Codes. Self-dual Codes and Invariant Theory. The Golay Codes. Association Schemes. Appendix A. Tables of the Best Codes Known. Appendix B. Finite Geometries. Bibliography. Index.