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Maximum scattered linear sets and MRD-codes

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Abstract

The rank of a scattered \({\mathbb F}_q\)-linear set of \({{\mathrm{{PG}}}}(r-1,q^n)\), rn even, is at most rn / 2 as it was proved by Blokhuis and Lavrauw. Existence results and explicit constructions were given for infinitely many values of r, n, q (rn even) for scattered \({\mathbb F}_q\)-linear sets of rank rn / 2. In this paper, we prove that the bound rn / 2 is sharp also in the remaining open cases. Recently Sheekey proved that scattered \({\mathbb F}_q\)-linear sets of \({{\mathrm{{PG}}}}(1,q^n)\) of maximum rank n yield \({\mathbb F}_q\)-linear MRD-codes with dimension 2n and minimum distance \(n-1\). We generalize this result and show that scattered \({\mathbb F}_q\)-linear sets of \({{\mathrm{{PG}}}}(r-1,q^n)\) of maximum rank rn / 2 yield \({\mathbb F}_q\)-linear MRD-codes with dimension rn and minimum distance \(n-1\).

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università degli Studi della Campania “Luigi Vanvitelli”, 81100, Caserta, Italy

    Bence Csajbók, Giuseppe Marino, Olga Polverino & Ferdinando Zullo

  2. Department of Geometry, MTA-ELTE Geometric and Algebraic Combinatorics Research Group, ELTE Eötvös Loránd University, Pázmány Péter Sétány 1/C, 1117, Budapest, Hungary

    Bence Csajbók

Authors
  1. Bence Csajbók
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  2. Giuseppe Marino
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  3. Olga Polverino
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  4. Ferdinando Zullo
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Corresponding author

Correspondence to Olga Polverino.

Additional information

The research was supported by Ministry for Education, University and Research of Italy MIUR (Project PRIN 2012 “Geometrie di Galois e strutture di incidenza”) and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA—INdAM).

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Csajbók, B., Marino, G., Polverino, O. et al. Maximum scattered linear sets and MRD-codes. J Algebr Comb 46, 517–531 (2017). https://doi.org/10.1007/s10801-017-0762-6

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  • DOI: https://doi.org/10.1007/s10801-017-0762-6

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