On Lambda-designs.

PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/179981/2/7004120.pd

On properties and constructions of t -designs, lambda -designs and perpendicular arrays

Combinatorial configurations have significant applications in many areas, such as in the design of experiments, coding theory, computer architecture, and information storage and retrieval. In this work, we will concentrate on three types of combinatorial configurations: t-designs, perpendicular arrays, and $\lambda$-designs. On t-designs. Khosrovshahi and Ajoodani-Namini give a new method for extending t-designs with k = t + 1. They obtain a recursive construction for t-designs and for large sets. Here we generalize their results to the general case k $\ge$ t + 1, and construct a family of large sets of $3 - (v,5,{v-3\choose 2}/3)$ designs with $v = 9m + 4 (m = 1,2,3,\...).$ Further we show that there exists a large set of $4-(9m + 5,6,{9m+1\choose 2}/3)$ designs for any m $\u3e$ 1 if there is a large set of 4$-$(13,5,3) designs. E. Kohler defines a new class of intersection numbers for t-designs, and obtains a formula relating them. As a result, he shows that certain t-designs do not exist. We generalize his intersection numbers and obtain a similar formula with expected applications. On perpendicular arrays. We show that there exists a t $-$ (v,t + 1,$\lambda\prime$) design if and only if there exists a PA\sb\lambda(t,t + 1,v) with $\lambda = {\lambda\prime\over (\lambda\prime,t + 1)}.$ Consequently, perpendicular arrays exist for all integers t $\u3e$ 0 and $\lambda$ = 1. Also if v $\not\equiv$ 0 (mod 3) then there exists a PA\sb1(3,4,v). Further, there is a PA\sb3(3,4,v) for every v $\ge$ 4. We exhibit several other infinite families of PA\u27s with t $\ge$ 3 and relatively small $\lambda$. We also discuss methods of constructing PA\u27s based on automorphism groups. These methods allow the construction of PA\u27s with k $\u3e$ t + 1. On $\lambda$-designs. So far, all known $\lambda$-designs are of type-1. The main goal on $\lambda$-designs is to prove Woodall\u27s conjecture that all $\lambda$-designs are type-1. Although there is ample evidence to support this conjecture, this problem seems very difficult and is still open. In this work we provide additional evidence in support of Woodall\u27s conjecture. Specifically, we shall give various formulae on the block intersection numbers of a $\lambda$-design; we improve the upper bound on v; and finally we show that Woodall\u27s conjecture is true in some additional situations