Question

(a) A consumer organization wishes to test 12 different new perfumes, and has devised a number of tests to accomplish this. These tests are subject the following constraints:

• only six perfumes are to be tested by each tester;

• each pair of perfumes is to be compared the same number of times.

(i) Explain how a block design can be used to set up this experiment.

(ii) Find the set of parameters (v, b, k, r, λ) of such a design where the number of testers is as small as possible. List one other appropriate set of parameters for such a design.

(iii) Use a construction (from those described in Design 4) to construct such a design for which the number of testers is as small as possible.

(b) Let Δ be a BIBD with k = 5 and λ = 2. Explain why the number of varieties in Δ must have either the form 10n + 1 or the form 10n + 5, for some integer n.

(c) (i) Is the design that you constructed in part (a)(iii) resolvable?

(ii) Is a design as in part (b) with v = 21 resolvable?

Explain your answers briefly.

(d) Let Δ be a BIBD, let Δ^_ be its complement, and let C and C^_ be the codes whose codcwords are the rows of the incidence matrices of Δ and Δ^_, respectively.

Show that the codes C and C^- have the same Hamming distance.