Vector space partitions and designs Part II – constructions


This article is the second part and companion article to Part I on the basictheory of what are called focal-spreads; partitions of finite vector spacesof dimension t+k by one subspace of dimension t (the `focus') and theremaining subspaces of dimension k, a `focal-spread of type (t,k)'. InPart I, additive focal-spreads are shown to be equivalent to additivepartial spreads. Focal-spreads of type (k+1,k) also produce %2-(q^{k+1},q,1)-designs, \ and various other double and triple-spreads.Also, in Part I, there are two different methods given to constructfocal-spreads, one of which is due to Beutelspacher, the other method beinga coordinate method similar to the theory available for translation planes.Here, we shall give a new construction that we term "going up," which alsoallows a specification of certain subplanes of the focal-spread.

DOI Code: 10.1285/i15900932v30n2p101

Keywords: vector space partition; designs; focal-spread; going up

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