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Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes

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Abstract

A monomial-Cartesian code is an evaluation code defined by evaluating a set of monomials over a Cartesian product. It is a generalization of some families of codes in the literature, for instance toric codes, affine Cartesian codes, and J-affine variety codes. In this work we use the vanishing ideal of the Cartesian product to give a description of the dual of a monomial-Cartesian code. Then we use such description of the dual to prove the existence of quantum error correcting codes and MDS quantum error correcting codes. Finally we show that the direct product of monomial-Cartesian codes is a locally recoverable code with t-availability if at least t of the components are locally recoverable codes.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Cleveland State University, Cleveland, OH, USA

    Hiram H. López & Ivan Soprunov

  2. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Gretchen L. Matthews

Authors
  1. Hiram H. López
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  2. Gretchen L. Matthews
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  3. Ivan Soprunov
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Correspondence to Hiram H. López.

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Gretchen L. Matthews was supported by NSF DMS-1855136.

This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Codes, Cryptology and Curves”.

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López, H.H., Matthews, G.L. & Soprunov, I. Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes. Des. Codes Cryptogr. 88, 1673–1685 (2020). https://doi.org/10.1007/s10623-020-00726-x

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  • DOI: https://doi.org/10.1007/s10623-020-00726-x

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