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

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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• Good Quantum Error-Correcting Codes Exist
• Multiple Particle Interference and Quantum Error Correction
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• LCD codes over ${\mathbb{F}}_q$ are as good as linear codes for q at least four
• Affine cartesian codes
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• Projective nested cartesian codes
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Cited by in corpus (7)

• Polar decreasing monomial-Cartesian codes
• On Steane-Enlargement of Quantum Codes from Cartesian Product Point Sets
• Optimal $(r,\delta )$-LRCs from monomial-Cartesian codes and their subfield-subcodes
• MDS, Hermitian Almost MDS, and Gilbert-Varshamov Quantum Codes from Generalized Monomial-Cartesian Codes
• Erasures repair for decreasing monomial-Cartesian and augmented Reed-Muller codes of high rate
• Multivariate Goppa codes
• The dual of an evaluation code