Fibonacci Designs

A Metis design is one for which v = r + k + 1. This paper deals with Metis designs that are quasi-residual. The parameters of such designs and the corresponding symmetric designs can be expressed by Fibonacci numbers. Although the question of existence seems intractable because of the size of the designs, the nonexistence of corresponding di¤erence sets can be dealt with in a substantive way. We also recall some inequalities for the number of …xed points of an automporphism of a symmetric design and suggest possible connections to the designs that would be the symmetric extensions of Metis designs. Key Words: quasi-symmetric design, symmetric design, Metis design, Fibonacci numbers, automorphism. 2010 Mathematics Subject Classi…cation: Primary: 05B05, 05B10; Secondary: 11B39 1. QUASI-RESIDUAL METIS DESIGNS In the paper [16] by McDonough, Mavron, and the author, a method of amalgamating nets and designs was presented that led to quasi-symmetric designs similar to those discovered by Bracken, McGuire, and the author [3]. At various stages of the construction, restrictions on the designs involved needed to be imposed in order to make the …nal amalgamation have desired regularity properties. One particular type of design was a generalization of Hadamard designs, and we named them Metis designs, in honor of Hadamard’s ancestral home, Metz. They are block designs whose standard parameter set (v; b; r; k; ) satis…es the additional relation v = r+ k+ 1. Symmetric Metis designs are indeed Hadamard designs. The family of Metis designs M has the following property: regard the parameter set for any design as a point in R on the variety D de…ned by the two standard design relations vr = bk and r(k 1) = (v 1). Then if a design belongs toM, there is a line in D through the corresponding point such that all the points on that line belong to M. There are other common families of designs with such a linear property. The nature of lines in D has been explored in the somewhat speculative preprint [22], and Section 4 describes some of the results. It seems a natural question to ask for Metis designs that are also quasi-residual. The parameters would satisfy the two equations v = r + k + 1 r = k + ;