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Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication

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Titel: Self-Orthogonal Codes Constructed from Posets and Their Applications in Quantum Communication
Autoren: Yansheng Wu, Yoonjin Lee
Quelle: Mathematics ; Volume 8 ; Issue 9 ; Pages: 1495
Verlagsinformationen: Multidisciplinary Digital Publishing Institute
Publikationsjahr: 2020
Bestand: MDPI Open Access Publishing
Schlagwörter: binary linear code, poset, weight distribution, self-orthogonal code, quantum code
Beschreibung: It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.
Publikationsart: text
Dateibeschreibung: application/pdf
Sprache: English
Relation: E1: Mathematics and Computer Science; https://dx.doi.org/10.3390/math8091495
DOI: 10.3390/math8091495
Verfügbarkeit: https://doi.org/10.3390/math8091495
Rights: https://creativecommons.org/licenses/by/4.0/
Dokumentencode: edsbas.403C7CF1
Datenbank: BASE
Beschreibung
Abstract:It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.
DOI:10.3390/math8091495