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Evasive subspaces
Author(s) -
Bartoli Daniele,
Csajbók Bence,
Marino Giuseppe,
Trombetti Rocco
Publication year - 2021
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21783
Subject(s) - linear subspace , mathematics , dimension (graph theory) , subspace topology , vector space , duality (order theory) , finite field , space (punctuation) , field (mathematics) , pure mathematics , combinatorics , discrete mathematics , mathematical analysis , computer science , operating system
Abstract Let V denote an r ‐dimensional vector space over F q n , the finite field of q n elements. Then V is also an r n ‐dimension vector space over F q . An F q ‐subspace U of V is( h , k ) q ‐evasive if it meets the h ‐dimensional F q n ‐subspaces of V in F 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 ⌊ r n ∕ 2 ⌋ when r n 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.