(a) Consider the following pair of 4 x 4 latin squares, X and Y, and
the 4 x 4 rectangular array A.
X:
0 1 2 3
1032
2301
3210
Y:
0231
3102
1320
2013
A
abcd
efgh
mnop
(i) Show that the latin squares X and Y are orthogonal.
(ii) A 4-replicate resolvable (0.1)-design A with 16 varieties is
constructed using the array A and the orthogonal latin
squares X and Y.
Find the parameters b, k and r for this design A.
(iii) The 4 x 4 latin square Z, with first row 0 3 1 2, is orthogonal
to both X and Y.
Explain briefly why the 0 entries have to appear where they
do in Z, then write down Z.
(iv) Use Z to construct a fifth replicate that extends design A.
(v) By considering the average concurrence, or otherwise, explain.
briefly why this extended 5-replicate design must be
balanced.
Question 14 DESIGN
a) Consider the following pair of 4 4 latin squares.X and Y.and the 4X4 rectangular array A.
0123 1032 2301 3210
0231 310 2 132 0 2013
a b e f 9 h A: i k 1
X:
Y:
m n
a p
() Show that the latin squares X and Y are orthogonal.
iiA 4-replicate resolvable 0.1)-design with 16 varieties is constructed using the array A and the orthogonal latin squares X and Y. Find the parameters band r for this design . iii The 4 4 latin square Z.with first row 03 12,is orthogonal to both X andY. Explain briefly why the 0 entries have to appear where they do in Z.then write down Z.
iv)Use Z to construct a fifth replicate that extends design .
vBy considering the average concurrence,or otherwise,explain briefly why this extended 5-replicate design must be balanced.
b Consider the following 7 x 7 latin square L,obtained by using the Steiner triple svstem construction with a Steiner triple system S
5 4 7 6
3 2 1 7 4 5 4 2 1 3 6 7 4 5 5 7 6 4 1 3 2 4 6 7 1 5 2 3
L:
7 5 4 3 2 6 1 6 4 5 2 3 1 7
(i) List the blocks of the Steiner triple system S
ii) Explain why it is not possible to start with a different Steiner triple system and use the Steiner triple system construction to construct a 7 7 latin square that is orthogonal to L
