Mutually quasi-orthogonal Latin squares.
This thesis considers problems concerning Latin squares and sets of mutually quasiorthogonal
Latin squares (MQOLS). We show how MQOLS are related to a number of
other designs and establish bounds on Nq(n), the maximum number of Latin squares
of order n in a mutually quasi-orthogonal set.
We report the number of quasi-complete mappings admitted by each group of
order 15 or less, and explain the surprising result that each of the non-cyclic groups
of order 8 possesses exactly 384 complete mappings. For each group G of order 15 or
less, we identify the sizes of all maximal sets of mutually orthogonal orthomorphisms.
We also identify a number of new maximal sets for larger groups.
We present a method to determine all proper, maximal sets of MQOLS of order
n and carry this out for n < 6. Also we present a search for 3 MQOLS of order 10,
which, whilst not identifying such a set, led to the identification of all resolutions of
each (10,3, 2)-balanced incomplete block design. We give a construction for MQOLS
based on groups, and use this to determine new sets of 2n - 1 MQOLS of order 2n
based on two infinite classes of group. Existence results for MQOLS based on groups
are also extended.
Two constructions for (n x n)/k semi-Latin squares are given, one of which provides
some new A-, D- and E-optimal examples with k > n which out-perform the
existing A-, D- and E-optimal examples in the E'-criteria. Finally we consider the
problem of determining invertible directed terraces of each non-abelian group of order
< 21, and in so doing construct the first doubly balanced bipartite tournament of odd