2026-04-14T00:31:13Zhttps://uvadoc.uva.es/oai/requestoai:uvadoc.uva.es:10324/646362024-01-17T20:02:18Zcom_10324_32197com_10324_952com_10324_894col_10324_32199
UVaDOC
author
Giménez, Philippe Thierry
author
Ruano Benito, Diego
author
San José Rubio, Rodrigo
2024-01-17T09:43:48Z
2024-01-17T09:43:48Z
2024
Finite Fields and Their Applications, 2024, vol. 94, 102353
1071-5797
https://uvadoc.uva.es/handle/10324/64636
10.1016/j.ffa.2023.102353
102353
Finite Fields and Their Applications
94
Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we generalize the necessary tools to deal with this case as well: we obtain a universal Gröbner basis for the vanishing ideal of the set of standard representatives of the projective space and we show how to reduce any monomial with respect to this Gröbner basis. With respect to the parameters of these codes, by considering subfield subcodes of projective Reed-Muller codes we obtain long linear codes with good parameters over a small finite field.
eng
Attribution-NonCommercial-NoDerivatives 4.0 Internacional
Coding theory
Software engineering
Subfield subcodes of projective Reed-Muller codes
info:eu-repo/semantics/article
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
URL
https://uvadoc.uva.es/bitstream/10324/64636/1/Subfield-subcodes-of-projective.pdf
File
MD5
ddd0e827e1000ba0abff3f32db2d6d63
794746
application/pdf
Subfield-subcodes-of-projective.pdf