On three-designs of small order |
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Authors: | Haim Hanani Alan Hartman Earl S Kramer |
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Institution: | Department of Mathematics, Technion I.I.R., Haifa, Israel;IBM Scientific Research Centre, Computer Science Building, Technion City, Haifa 32000, Israel;Department of Mathematics, University of Nebraska, Lincoln, NE 68588, USA |
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Abstract: | For positive integers t?k?v and λ we define a t-design, denoted Bik,λ;v], to be a pair (X,B) where X is a set of points and B is a family, (Bi:i?I), of subsets of X, called blocks, which satisfy the following conditions: (i) |X|=v, the order of the design, (ii) |Bi|=k for each i?I, and (iii) every t-subset of X is contained in precisely λ blocks. The purpose of this paper is to investigate the existence of 3-designs with 3?k?v?32 and λ>0.Wilson has shown that there exists a constant N(t, k, v) such that designs Btk,λ;v] exist provided λ>N(t,k,v) and λ satisfies the trivial necessary conditions. We show that N(3,k,v)=0 for most of the cases under consideration and we give a numerical upper bound on N(3, k, v) for all 3?k?v?32. We give explicit constructions for all the designs needed. |
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