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Por favor, use este identificador para citar o enlazar este ítem: http://uvadoc.uva.es/handle/10324/31741
Título: The metric structure of linear codes
Autor: Ruano, Diego
Año del Documento: 2018
Editorial: Springer
Descripción: Producción Científica
Documento Fuente: The metric structure of linear codes. In Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, pages 537-561. Editors: G.-M. Greuel, L. Narváez Macarro, S. Xambó-Descamps. Springer Verlag. ISBN: 978-3-319-96826-1 (2018)
Resumen: The bilinear form with associated identity matrix is used in coding theory to define the dual code of a linear code, also it endows linear codes with a metric space structure. This metric structure was studied for generalized toric codes and a characteristic decomposition was obtained, which led to several applications as the construction of stabilizer quantum codes and LCD codes. In this work, we use the study of bilinear forms over a finite field to give a decomposition of an arbitrary linear code similar to the one obtained for generalized toric codes. Such a decomposition, called the geometric decomposition of a linear code, can be obtained in a constructive way; it allows us to express easily the dual code of a linear code and provides a method to construct stabilizer quantum codes, LCD codes and in some cases, a method to estimate their minimum distance. The proofs for characteristic 2 are different, but they are developed in parallel.
Patrocinador: The author gratefully acknowledges the support from RYC-2016-20208 (AEI/FSE/UE), the support from The Danish Council for Independent Research (Grant No. DFF-4002-00367), and the support from the Spanish MINECO/FEDER (Grants No. MTM2015-65764-C3-2-P and MTM2015-69138-REDT).
Version del Editor: https://doi.org/10.1007/978-3-319-96827-8_24
Idioma: eng
URI: http://uvadoc.uva.es/handle/10324/31741
Derechos: info:eu-repo/semantics/openAccess
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