Embedding of a pseudo-point residual design into a Möbius plane

✍ Scribed by Agnes Hui Chan

• Book ID: 103057056
• Publisher: Elsevier Science
• Year: 1981
• Tongue: English
• Weight: 462 KB
• Volume: 37
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(81)90137-0

✦ Synopsis

Let % be a class of subsets of a finite set X. Elements of $!I are called blocks. Let u, t, A and k be nonnegative integers such thar u B k 3 t 3 0. A pair (X, '%?l) is called a (u, k, A) t-design, denoted by S,(t, k, u), if (1) 1x1 = u, (2) every t-subset of X is contained in exactly A blocks and (3) for every block A in 8, IAl = k. A Mobius plane M is an S,(3, q + 1, q* + 1) where q is a positive integer. Let 30 be a fixed point in M. If 00 is deleted ficqm M, t0gethe.r with all the blocks containing 00, then we obtain a point-residual design M*. It can be easily checked that M* is an S,(2, q + 1, q*). Any S,(2, q+ 1, q2) is called a pseudo-point-residual design of order q, abbreviated by PPRD(q). Let A and I3 be two blocks ill E PPRD(q)M*. A and B are said to be tangent to each other at z if and only if A nZ3 ={z}. &4* is sa:d to have the Tangency Ropcrty if for any block A in M*, and points x and y such that x E 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)w is uniquely embeddable into a Mobius plane if and only if M* satisfies the Tangency Property.