ABSTRACT
We have introduced focal-spreads in the previous chapters and have noted that the known examples arise from k-cuts of translation planes. In this chapter, we provide some very general constructions of focalspreads that do not appear to be k-cuts, although this is still an open question. This process allows us to specify subspaces of order qk, in a focal-
spread of type (k(s1); k), where the focus has dimension k(s1) = t and we have an associated partial Sperner k-spread of qt k-dimensional subspaces. Furthermore, if a focal-spread is de…ned more generally as a partition of a vector space of dimension t + k over GF (q) by one subspace of dimension t0 and the remaining subspaces of dimension k, we say the focal-spread is of type (t; t0; k), when t0 < t. The going up process allows constructions of this more general type of focal-spreads. If we begin with a vector space Vtk of dimension tk-over GF (q),
we may construct a k-spread in Vtk by a choice of any sequence of t2P j=0
(t j 1) = Nt translation planes of order qk and associated
spreads Si, for i = 1; 2; ::; Nt. The reader is directed to the article by Jha and Johnson [80] for
additional details. The construction that we give basically relies on another construc-
tion of Sperner spreads from suitably many k-spreads. In this construction, we again use the concept of j-(0-sets).
