Mathematics 502
Homework (due Mar 7)
A. Hulpke
19) For n > 2 let X be the points of PGn (q) and B the i-flats (i.e. the i + 1 dimensional
subspaces of GF(q)n+1 ). What are the parameters of such a design?
20) a) Why can’t there be a 4 − (8, 6, 1) design?
b) Suppose that k − 1 is a prime. Show that there are 2 − (v, k, λ) designs only if k − 1 | v − 1
or k − 1 | λ.
21) An extension of a t − (v, k, λ) design (X , B ) is a (t + 1) − (v + 1, k + 1, λ) design (Y, C )
with a point y such that its derived design with respect to y is isomorphic to X. Prove
that a necessary condition for a t − (v, k, λ) design with bb blocks to have an extension is
that v + 1 divides b(k + 1). Hence show that, if a projective plane of order q > 1 has an
extension, then q ∈ {2, 4, 10}.
22) Consider 15 points, partitioned into a set X = {x0 , . . . , x6 } of 7 points and a second
set Y = { y0 , . . . , y7 } of 8 points. We now form blocks of size 3 on X ∪ Y:
1. On X1 take the block structure given by PG2 (2).
2. We also form 28 triples containing points of Y: For every i = 0, . . . , 6 consider the
sets {xi , yi , y7 } and for k = 1, 2, 3 the sets {xi , yi+k , yi−k } with arithmetic in the indices
done modulo 7.
a) Show that these 7 + 28 = 35 sets form a 2 − (15, 3, 1) design.
b) Solve Kirkman’s schoolgirl problem (T. P. Kirkman, Query VI. Lady’s and Gentleman’s
Diary (1850), 48):
Fifteen young ladies of a school walk out three abreast for seven days in succession: it is required to arrange them daily so that no two shall walk abreast
more than once.
23) Let J = 1n×n and I the identity matrix. Show that
det(xI + yJ) = (x + yn)xn−1 .
(Hint: The matrix is symmetric and thus has can be diagonalized, eigenspaces are mutually orthogonal)