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Projective Reed–Muller type codes on higher dimensional scrolls

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Abstract

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.

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Acknowledgements

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. Faculdade de Matemática, Universidade Federal de Uberlândia, Av. J. N. Ávila 2121, 38.408-902, Uberlândia, MG, Brazil

    Cícero Carvalho & Victor G. L. Neumann

  2. Departamento de Matemáticas, Universidad Autónoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, C.P. 09340, Delegación Iztapalapa, CD. Mexico, Mexico

    Xavier Ramírez-Mondragón & Horacio Tapia-Recillas

Authors
  1. Cícero Carvalho
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  2. Xavier Ramírez-Mondragón
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  3. Victor G. L. Neumann
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  4. Horacio Tapia-Recillas
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Corresponding author

Correspondence to Xavier Ramírez-Mondragón.

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Communicated by G. Korchmaros.

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C. Carvalho and V. G. L. Neumann are partially supported by CNPq and FAPEMIG. X. Ramírez-Mondragón is supported by CONACyT Scholarship No. 209918.

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Carvalho, C., Ramírez-Mondragón, X., Neumann, V.G.L. et al. Projective Reed–Muller type codes on higher dimensional scrolls. Des. Codes Cryptogr. 87, 2027–2042 (2019). https://doi.org/10.1007/s10623-018-00603-8

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  • DOI: https://doi.org/10.1007/s10623-018-00603-8

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