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Báo cáo toán học: "On the most Weight w Vectors in a Dimension k Binary Code"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài:On the most Weight w Vectors in a Dimension k Binary Code. | On the most Weight w Vectors in a Dimension k Binary Code Joshua Brown Kramer Department of Mathematics and Computer Science Illinois Wesleyan University jbrownkr@iwu.edu Submitted Jan 17 2009 Accepted Aug 10 2010 Published Oct 29 2010 Mathematics Subject Classifications 05D05 05E99 Abstract Ahlswede Aydinian and Khachatrian posed the following problem what is the maximum number of Hamming weight w vectors in a k-dimensional subspace of Fn The answer to this question could be relevant to coding theory since it sheds light on the weight distributions of binary linear codes. We give some partial results. We also provide a conjecture for the complete solution when w is odd as well as for the case k 2w and w even. One tool used to study this problem is a linear map that decreases the weight of nonzero vectors by a constant. We characterize such maps. 1 Introduction Ahlswede Aydinian and Khachatrian 1 introduced extremal problems with dimension constraints. Begin with a class of set systems on the ground set n 1 2 . n . For example the set of intersecting families on n . Given a field F a set system in this class can be viewed as a collection of 0 1 -valued vectors in Fn. The extremal problem with a dimension constraint is to find the largest set system that has rank at most k. In this paper we consider a dimension constraint on uniform hypergraphs. To be more precise first recall that the Hamming weight of a vector v denoted wt v is the number of entries of v that are nonzero. Given n k w G N and a field F denote MF n k w to be the maximum number of 0 1 -valued vectors with Hamming weight w in a k-dimensional subspace of Fn. Ahlswede Aydinian and Khachatrian found a formula for MR 1 . Theorem 1 Ahlswede Aydinian and Khachatrian . Given n k w G N MR n k w MR n k n w THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R142 1 and for w n 2 kÌ if 2w k WZ MRn k w I f- 22w-k if k 2w 2 k - 1 2k if k - 1 w. This paper focusses on the case F F2. Given n k w G N denote m n k w MF2 n