Almost Affinely Disjoint Subspaces

07/03/2020

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by   Hedongliang Liu, et al.

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In this work, we introduce a natural notion concerning vector finite spaces. A family of k-dimensional subspaces of 𝔽_q^n is called almost affinely disjoint if any (k+1)-dimensional subspace containing a subspace from the family non-trivially intersects with only a few subspaces from the family. The central question discussed in the paper is the polynomial growth (in q) of the maximal cardinality of these families given the parameters k and n. For the cases k=1 and k=2, optimal families are constructed. For other settings, we find lower and upper bounds on the polynomial growth. Additionally, some connections with problems in coding theory are shown.