5 research outputs found
A lower bound on HMOLS with equal sized holes
It is known that , the maximum number of mutually orthogonal latin
squares of order , satisfies the lower bound for large
. For , relatively little is known about the quantity ,
which denotes the maximum number of `HMOLS' or mutually orthogonal latin
squares having a common equipartition into holes of a fixed size . We
generalize a difference matrix method that had been used previously for
explicit constructions of HMOLS. An estimate of R.M. Wilson on higher
cyclotomic numbers guarantees our construction succeeds in suitably large
finite fields. Feeding this into a generalized product construction, we are
able to establish the lower bound for any
and all
Existence of r-fold perfect (v,K,1)-Mendelsohn designs with K⊆{4,5,6,7}
AbstractLet v be a positive integer and let K be a set of positive integers. A (v,K,1)-Mendelsohn design, which we denote briefly by (v,K,1)-MD, is a pair (X,B) where X is a v-set (of points) and B is a collection of cyclically ordered subsets of X (called blocks) with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B. If for all t=1,2,…,r, every ordered pair of points of X are t-apart in exactly one block of B, then the (v,K,1)-MD is called an r-fold perfect design and denoted briefly by an r-fold perfect (v,K,1)-MD. If K={k} and r=k−1, then an r-fold perfect (v,{k},1)-MD is essentially the more familiar (v,k,1)-perfect Mendelsohn design, which is briefly denoted by (v,k,1)-PMD. In this paper, we investigate the existence of r-fold perfect (v,K,1)-Mendelsohn designs for a specified set K which is a subset of {4, 5, 6, 7} containing precisely two elements