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Book spreads in \(\mathrm{PG}(7,2)\). (English) Zbl 1295.51013

Summary: An \((n,q,r,s)\) book is a collection of \(r\)-subspaces in \(\mathrm{PG}(n,q)\) called pages, which cover the whole projective space and intersect in a common \(s\)-subspace called the spine such that any point outside the spine is in exactly one page. An \((n,q,r,s)\) book \(t\)-spread is a \(t\)-spread in \(\mathrm{PG}(n,q)\) for which there exists an \((n,q,r,s)\) book, such that the points of each page of this book and hence the points of the spine are partitioned by \(t\)-subspaces of the \(t\)-spread. We commence by showing that an \((n,q,r,s)\) book \(t\)-spread exists if and only if the following three conditions hold: \[ \text{(ii) } (r-s)| (n-s),\text{(ii) }(t+1)| (s+1),\text{(iii) }(t+1)| (r+1). \] In general, the number of different kinds of \((n,q,r,s)\) book \(t\)-spreads is a tiny proportion of the number of different kinds of \(t\)-spreads in \(\mathrm{PG}(n,q)\). In the rest of this paper we present computer-aided classification results for certain types of \((7,2,5,3)\) book 1-spreads.

MSC:

51E21 Blocking sets, ovals, \(k\)-arcs
Full Text: DOI

References:

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