Projective Reed–Muller type codes on higher dimensional scrolls

  • Cícero Carvalho
  • Xavier Ramírez-MondragónEmail author
  • Victor G. L. Neumann
  • Horacio Tapia-Recillas


In 1988 Lachaud introduced the class of projective Reed–Muller codes, defined by evaluating the space of homogeneous polynomials of a fixed degree d on the points of \(\mathbb {P}^n(\mathbb {F}_q)\). In this paper we evaluate the same space of polynomials on the points of a higher dimensional scroll, defined from a set of rational normal curves contained in complementary linear subspaces of a projective space. We determine a formula for the dimension of the codes, and the exact value of the dimension and the minimum distance in some special cases.


Projective variety codes Evaluation codes Reed–Muller type codes Higher dimensional scroll 

Mathematics Subject Classification

11T71 13P25 94B60 



The second author is grateful to Prof. F. Zaldivar for pointing out the concept of a higher dimensional scroll. We thank the referees for a careful reading and their comments which improved the manuscript.


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Authors and Affiliations

  1. 1.Faculdade de MatemáticaUniversidade Federal de UberlândiaUberlândiaBrazil
  2. 2.Departamento de MatemáticasUniversidad Autónoma MetropolitanaDelegación IztapalapaMexico

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