Embedding of a pseudo-point residual design into a Möbius plane |
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Authors: | Agnes Hui Chan |
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Institution: | Department of Mathematics, Northeastern University, Boston, MA 02115, USA |
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Abstract: | Let be a class of subsets of a finite set X. Elements of are called blocks. Let υ, t, λ and k be nonnegative integers such that υ?k?t?0. A pair (X, ) is called a (υ, k, λ) t-design, denoted by Sλ(t, k, υ), if (1) |X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in , |A| = k. A Möbius plane M is an S1(3, q+1, q2+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M*. It can be easily checked that M* is an Sq(2, q+1, q2). Any Sq(2, q+1, q2) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and B be two blocks in a PPRD(q)M*. A and B are said to be tangent to each other at z if and only if A∩B={z}. M* is said to have the Tangency Property if for any block A in M*, and points x and y such that x?A and y?A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD(q)M* is uniquely embeddable into a Möbius plane if and only if M* satisfies the Tangency Property. |
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