For a partition \(\mu \) of a positive integer and a prime \(p\) , let \(A_{(p, \mu )}\) be a finite Abelian \(p\) -group, and let \(\widehat{A}_{(p, \mu )}\) be its dual group. We define a finite Abelian group as \(G = \bigoplus A_{(p, \mu )}\) and its dual as \(\widehat{G} = \bigoplus \widehat{A}_{(p, \mu )}\) . In this paper, we explore the symplectic structure associated with the group \(Z_{(p, \mu )} = A_{(p, \mu )} \oplus \widehat{A}_{(p, \mu )}\) 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 \(Z_{(p, \mu )}\) . Additionally, we present a study of \(\mu \) -based poset orbit codes and the operators of the algebra \(Hom_{\mathbb {C}}(L^2(G), L^2(G))\) . This interaction among the order ideals of a poset, \(\mu \) -based poset orbit codes, group symmetries, and the operators in the algebra \(Hom_{\mathbb {C}}(L^2(G), L^2(G))\) 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.
