TAILIEUCHUNG - Báo cáo toán học: "The Fraction of Subspaces of GF(q)n with a Specified Number of Minimal Weight Vectors is Asymptotically Poisson"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The Fraction of Subspaces of GF(q)n with a Specified Number of Minimal Weight Vectors is Asymptotically Poisson . | The Fraction of Subspaces of GF q n with a Specihed Number of Minimal Weight Vectors is Asymptotically Poisson Edward A. Bender Center for Communications Research 4320 Westerra Court San Diego CA 92121 USA ed@ E. Rodney Canheld Department of Computer Science University of Georgia Athens GA 30602 USA erc@ Submitted August 30 1996 Accepted November 27 1996 Abstract The weight of a vector in the hnite vector space GF q n is the number of nonzero components it contains. We show that for a certain range of parameters n j k w the number of k-dimensional subspaces having j q 1 vectors of minimum weight w has asymptotically a Poisson distribution with parameter A n q 1 w 1qk n. As the Poisson parameter grows the distribution becomes normal. AMS-MOS Subject Classihcation 1990 . Primary 05A16 Secondary 05A15 11T99 THE ELECTRONIC JOURNAL OF COMBINATORICS 4 1997 R3 2 1. Introduction Almost all the familiar concepts of linear algebra such as dimension and linear independence are valid without regard to the characteristic of the underlying held. An example of a characteristic-dependent result is that a nonzero vector cannot be orthogonal to itself researchers accustomed to real vector spaces must modify their intuition on this point when entering the realm of hnite helds. Let q be a prime power hxed for the remainder of the paper and GF q be the hnite held with q elements. Because the underlying held is hnite there are many counting problems associated with fundamental concepts of linear algebra for example how many subspaces of dimension k are there in the vector space GF q n The answer is often denoted ln q and we have nl _ 1 - qn 1 - qn-1 1 - qn-k 1 k q 1 - qk 1 - qk-1 1 - q the Gaussian polynomial. The reader may consult 2 for an introduction to the subject. Dehne the weight of a vector v in GF q n to be the number of nonzero coordinates in v. The interaction of weight with familiar concepts of linear algebra yields more and harder counting problems. .

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