On-orbit codes of posets and operators in algebra - Mesnager, Sihem; Raja, Rameez (Journal of algebraic combinatorics, 2025)
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.