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Báo cáo toán học: "Codes, Lattices, and Steiner Systems"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Codes, Lattices, and Steiner Systems. | Codes Lattices and Steiner Systems Patrick Sole CNRS I3S ESSI BP 145 Route des Colles 06 903 Sophia Antipolis France Submitted February 16 1996 Accepted January 31 1997 Abstract Two classification schemes for Steiner triple systems on 15 points have been proposed recently one based on the binary code spanned by the blocks the other on the root system attached to the lattice affinely generated by the blocks. It is shown here that the two approaches are equivalent. 1991 AMS Classification Primary 05B07 Secondary 11H06 94B25. 1 Introduction It has been known since 1919 1919 that there are 80 Steiner triple systems on 15 points. Recently two algebraic invariants have been proposed to classify them. Let V denote the 35 block vectors vi of length 15 and hamming weight 3 of such a system. One can attach to V either the binary linear code C spanned by the vectors of V TW sole@alto.unice.fr 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 4 1997 R6 2 the lattice L Pj ZịVi Pị Zị 0 Zị 2 Zg DG The lattice L has norm 2 and its norm 2 vectors afford a possibly empty root system R. It so happens that exactly 5 non-equivalent codes C and also 5 non-equivalent root systems R occur and that they induce the same partition of the 80 S 2 3 15 in five parts. We shall provide a conceptual explanation of this experimental fact. 2 Notations and Definitions A Steiner triple system S 2 3 v is a 2 v 3 1 design. A binary code of length n and dimension k is a k dimensional vector subspace of F . The Hamming weight of a vector of F2 is the number of non-zero coordinates it contains. An n dimensional lattice is a discrete Z module of Rra which may or may not be of maximal rank n. The squared euclidean norm of a vector x of Rra is x.x. The norm of a lattice is the minimum nonzero norm of its elements. A lattice is integral if the dot product of any two of its vectors is an integer. An integral lattice is called even or type II in SPLAG if the norm of each its vectors is an even integer. A root in an .