On-orbit codes of posets and operators in algebra

For a partition μ of a positive integer and a prime p , let A ( p , μ ) be a finite Abelian p -group, and let A ^ ( p , μ ) be its dual group. We define a finite Abelian group as G = ⨁ A ( p , μ ) and its dual as G ^ = ⨁ A ^ ( p , μ ) . In this paper, we explore the symplectic structure associated w...

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Published in:	Journal of algebraic combinatorics Vol. 62; no. 3; p. 44
Main Authors:	Mesnager, Sihem, Raja, Rameez
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.11.2025 Springer Nature B.V
Subjects:	Algebra Automorphisms Codes Coding theory Combinatorial analysis Combinatorics Computer Science Convex and Discrete Geometry Cryptography Decomposition Error correction & detection Fourier transforms Group theory Group Theory and Generalizations Hilbert space Information theory Lattices Mathematics Mathematics and Statistics Operators (mathematics) Orbits Order Ordered Algebraic Structures Quantum phenomena Quantum physics 05E10 08A35 06A11 Automorphism Orbit Code Poset Hilbert space Representation Finite Abelian Group 81P70
ISSN:	0925-9899, 1572-9192
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Abstract	For a partition μ of a positive integer and a prime p , let A ( p , μ ) be a finite Abelian p -group, and let A ^ ( p , μ ) be its dual group. We define a finite Abelian group as G = ⨁ A ( p , μ ) and its dual as G ^ = ⨁ A ^ ( p , μ ) . In this paper, we explore the symplectic structure associated with the group Z ( p , μ ) = A ( p , μ ) ⊕ A ^ ( p , μ ) and consider its action on the Hilbert space L 2 ( G ) . We investigate the correspondence between the ideals of a poset, which represent the sizes of bit strings, and the lengths of automorphism orbit code words . We also examine the orbits that result from the action of the symplectic structure on the group Z ( p , μ ) . Additionally, we present a study of μ -based poset orbit codes and the operators of the algebra H o m C ( L 2 ( G ) , L 2 ( G ) ) . This interaction among the order ideals of a poset, μ -based poset orbit codes, group symmetries, and the operators in the algebra H o m C ( L 2 ( G ) , L 2 ( G ) ) bridges the gap between combinatorial coding theory and quantum systems. It also provides practical insights for constructing quantum protocols in the context of modern quantum information theory and cryptography.
AbstractList	For a partition μ of a positive integer and a prime p , let A ( p , μ ) be a finite Abelian p -group, and let A ^ ( p , μ ) be its dual group. We define a finite Abelian group as G = ⨁ A ( p , μ ) and its dual as G ^ = ⨁ A ^ ( p , μ ) . In this paper, we explore the symplectic structure associated with the group Z ( p , μ ) = A ( p , μ ) ⊕ A ^ ( p , μ ) and consider its action on the Hilbert space L 2 ( G ) . We investigate the correspondence between the ideals of a poset, which represent the sizes of bit strings, and the lengths of automorphism orbit code words . We also examine the orbits that result from the action of the symplectic structure on the group Z ( p , μ ) . Additionally, we present a study of μ -based poset orbit codes and the operators of the algebra H o m C ( L 2 ( G ) , L 2 ( G ) ) . This interaction among the order ideals of a poset, μ -based poset orbit codes, group symmetries, and the operators in the algebra H o m C ( L 2 ( G ) , L 2 ( G ) ) bridges the gap between combinatorial coding theory and quantum systems. It also provides practical insights for constructing quantum protocols in the context of modern quantum information theory and cryptography. For a partition μ of a positive integer and a prime p, let A(p,μ) be a finite Abelian p-group, and let A^(p,μ) be its dual group. We define a finite Abelian group as G=⨁A(p,μ) and its dual as G^=⨁A^(p,μ). In this paper, we explore the symplectic structure associated with the group Z(p,μ)=A(p,μ)⊕A^(p,μ) and consider its action on the Hilbert space L2(G).We investigate the correspondence between the ideals of a poset, which represent the sizes of bit strings, and the lengths of automorphism orbit code words. We also examine the orbits that result from the action of the symplectic structure on the group Z(p,μ). Additionally, we present a study of μ-based poset orbit codes and the operators of the algebra HomC(L2(G),L2(G)). This interaction among the order ideals of a poset, μ-based poset orbit codes, group symmetries, and the operators in the algebra HomC(L2(G),L2(G)) bridges the gap between combinatorial coding theory and quantum systems. It also provides practical insights for constructing quantum protocols in the context of modern quantum information theory and cryptography.
ArticleNumber	44
Author	Mesnager, Sihem Raja, Rameez
Author_xml	– sequence: 1 givenname: Sihem orcidid: 0000-0003-4008-2031 surname: Mesnager fullname: Mesnager, Sihem organization: Department of Mathematics, University of Paris VIII, F-93526 Saint-Denis, Laboratory Analysis, Geometry and Applications, LAGA, University Sorbonne Paris Nord, CNRS, UMR 7539, Telecom Paris, Polytechnic institute of Paris – sequence: 2 givenname: Rameez surname: Raja fullname: Raja, Rameez email: rameeznaqash@nitsri.ac.in organization: Department of Mathematics, National Institute of Technology Srinagar
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Cites_doi	10.1016/j.laa.2003.09.001 10.1016/j.disc.2013.11.001 10.37236/4322 10.1016/j.disc.2024.113900 10.1016/j.ejc.2020.103228 10.1017/CBO9780511805967 10.1109/18.104333
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SubjectTerms	Algebra Automorphisms Codes Coding theory Combinatorial analysis Combinatorics Computer Science Convex and Discrete Geometry Cryptography Decomposition Error correction & detection Fourier transforms Group theory Group Theory and Generalizations Hilbert space Information theory Lattices Mathematics Mathematics and Statistics Operators (mathematics) Orbits Order Ordered Algebraic Structures Quantum phenomena Quantum physics
Title	On-orbit codes of posets and operators in algebra
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