Embedding of a pseudo-residual design into a Möbius plane

Abstract

AbstractLet U be a class of subsets of a finite set X. Elements of U are called blocks. Let v, t and λ1, 0 ⩽ i ⩽ t, be nonnegative integers, and K be a subset of nonnegative integers such that every member of K is at most v. A pair (X, U) is called a (λ0, λ1,…, λt; K, υ)t-design if (1) |X| = υ, (2) every i-subset of X is contained in exactly λt blocks, 0 ⩽ i ⩽ t, and (3) for every block A in U, |A| ϵ K. It is well-known that if K consists of a singleton k, then λ0,…, λt − 1 can be determined from υ, t, k and λt. Hence, we shall denote a (λ0,…, λt; {k}, υ)t-design by Sλ(t, k, υ), where λ = λt. A Möbius plane M is an S1(3, q + 1, q2 + 1), where q is a positive integer. Let A be a fixed block in M. If A is deleted from M together with the points contained in A, then we obtain a residual design M′ with parameters λ0 = q3 + q − 1, λ1 = q2 + q, λ2 = q + 1, λ3 = 1, K = {q + 1, q, q − 1}, and υ = q2 − 1. We define a design to be a pseudo-block-residual design of order q (abbreviated by PBRD(q)) if it has these parameters. We consider the reconstruction problem of a Möbius plane from a given PBRD(q). Let B and B′ be two blocks in a residual design M′. If B and B′ are tangent to each other at a point x, and there exists a block C of size q + 1 such that C is tangent to B at x and is secant to B′, then we say B is r-tangent to B′ at x. A PBRD(q) is said to satisfy the r-tangency condition if for every block B of size q, and any two points x and y not in B, there exists at most one block which is r-tangent to B and contains x and y. We show that any PBRD(q)D can be uniquely embedded into a Möbius plane if and only if D satisfies the r-tangency condition