New maximum scattered linear sets of the projective line

Csajbók, Bence and Marino, Giuseppe and Zullo, Ferdinando (2018) New maximum scattered linear sets of the projective line. Finite Fields and their Applications (54). pp. 133-150. ISSN 1071-5797

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Abstract

In [A. Blokhuis, M. Lavrauw: Scattered spaces with respect to a spread in PG(n,q), Geom. Dedicata 81 (2000), 231--243.] and [G. Lunardon, O. Polverino: Blocking Sets and Derivable Partial Spreads, J. Algebraic Combin. 14 (2001), 49-56.] are presented the first two families of maximum scattered GF(q)-linear sets of the projective line PG(1,q^n). More recently in [J. Sheekey: A new family of linear maximum rank distance codes, Adv. Math. Commun. 10(3) (2016), 475-488.] and in [B. Csajbók, G. Marino, O. Polverino, C. Zanella: A new family of MRD-codes, Linear Algebra Appl. 548 (2018), 203-220.], new examples of maximum scattered GF(q)-subspaces of V(2,q^n) have been constructed, but the equivalence problem of the corresponding linear sets is left open. Here we show that the GF(q)-linear sets presented in [J. Sheekey: A new family of linear maximum rank distance codes, Adv. Math. Commun. 10(3) (2016), 475-488.] and in [B. Csajbók, G. Marino, O. Polverino, C. Zanella: A new family of MRD-codes, Linear Algebra Appl. 548 (2018), 203-220.], for n=6,8, are new. Also, for q odd, q congruent to 1, -1, or 0 (mod 5), we present new examples of maximum scattered GF(q)-linear sets in PG(1,q^6), arising from trinomial polynomials, which define new GF(q)-linear MRD-codes of GF(q)^6×6 with dimension 12, minimum distance 5 and left idealiser isomorphic to GF(q^6).

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