Rank-metric codes, linear sets, and their duality

Abstract

In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of Sheekey (Adv Math Commun 10:475–488, 2016, Sect. 5) on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in Lunardon (J Comb Theory Ser A 149:1–20, 2017). Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line \(\mathrm {PG}(1,q^n)\).

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Correspondence to John Sheekey.

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Geertrui Van de Voorde was supported by the Marsden Fund Council administered by the Royal Society of New Zealand.

Communicated by G. Lunardon.

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Sheekey, J., Van de Voorde, G. Rank-metric codes, linear sets, and their duality. Des. Codes Cryptogr. 88, 655–675 (2020). https://doi.org/10.1007/s10623-019-00703-z

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Keywords

  • MRD code
  • Weight distribution
  • Linear set
  • Scattered with respect to hyperplanes

Mathematics Subject Classification

  • 51E20
  • 05B25
  • 94B27