Evasive subspaces

Bartoli, Daniele and Csajbók, Bence and Marino, Giuseppe and Trombetti, Rocco (2021) Evasive subspaces. Journal of Combinatorial Designs, 29 (8). pp. 533-551. ISSN 1063-8539

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Abstract

Let V denote an r-dimensional vector space over GF(q^n), the finite field of q^n elements. Then V is also an rn-dimension vector space over GF(q). A GF(q)-subspace U of V is (h,k)_q-evasive if it meets the h-dimensional GF(q^n)-subspaces of V in GF(q)-subspaces of dimension at most k. The (1,1)_q-evasive subspaces are known as scattered and they have been intensively studied in finite geometry, their maximum size has been proved to be the lower integer part of rn/2 when rn is even or n=3. We investigate the maximum size of (h,k)_q-evasive subspaces, study two duality relations among them and provide various constructions. In particular, we present the first examples, for infinitely many values of q, of maximum scattered subspaces when r=3 and n=5. We obtain these examples in characteristics 2, 3 and 5.

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