6533b7dbfe1ef96bd1271326

RESEARCH PRODUCT

Partial spreads in finite projective spaces and partial designs

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A partial t-spread of a projective space P is a collection 5 p of t-dimensional subspaces of P of the same order with the property that any point of P is contained in at most one element of 50. A partial t-spread 5 p of P is said to be a t-spread if each point of P is contained in an element of 5P; a partial t-spread which is not a spread will be called strictly partial. Partial t-spreads are frequently used for constructions of affine planes, nets, and Sperner spaces (see for instance Bruck and Bose [5], Barlotti and Cofman [2]). The extension of nets to affine planes is related to the following problem: When can a partial t-spread 5 ~ of a projective space P be embedded into a larger partial t-spread Y ' of P? A strictly partial t-spread ~ which cannot be embedded into any partial t-spread 5 p' of the same projective space as a proper subset will be called a maximal strictly partial t-spread (or, shortly a rasp t-spread). Mesner [8] and Bruen [6] have proved that if 15 el denotes the cardinality of a rasp 1-spread 5 P of a three-dimensional projective space of finite order q, then