Homework 8
Math 413
Due Friday April 26 at 8pm
Bonus 10% for solutions typeset with LaTex
Show your work for full credit.
For this assignment, please submit your choice of 5 questions.
Question 1. Let R be a BIBD with parameters b,v,k,r,λ whose set of varieties is X and whose blocks are B1 ,..., Bb. Define Rc , the complementary design, whose blocks are
B 1(c),..., Bb(c), i.e., the complements of Bb in the set X. Show that Rc is a BIBD and calculate
its corresponding parameters b′ , v′ , k′ , r′ ,λ′ .
Question 2. Find a difference set in Z11 of size 5. What is the index (λ) that the corresponding SBIBD has?
Question 3. Extend the observation in class on Latin square constructions.
Let r be relatively prime to n. Define an n xn array A by
aij = ri + j (mod n), for i,j e Zn.
Show that A is a Latin square.
Question 4. Construct a completion of the 3 x 6 Latin rectangle.
Question 5. A Latin square is idempotent if its entries on the long diagonal running upper left to lower right are 0, 1,..., n — 1. A Latin square A is symmetric if aij = aji.
(a) Show there exists a symmetric idempotent n x n Latin square for all odd n.
(b) Show that if A is a symmetric idempotent n x n Latin square, then n is odd.
Question 6. Let n ≥ 2 be an integer.
(a) Prove that an n × (n − 2) Latin rectangle has at least two completions. Note if there is one completion, then by swapping the order of the final two rows, there must be another completion.
(b) For each n, find an example of ann×(n−2) Latin rectangle with exactly two completions.