Classical and Quantum MacWilliams Transforms as Spin Kinematics

Spin is the hidden engine behind the zoo of MacWilliams transforms in weight enumerator theories - not only for qubits and qudits, but even for classical codes. From nothing more than a split into trivial and nontrivial errors, we kinematically derive the MacWilliams transform as a Wigner-$D$ rotation between two canonical bases. Within each classical and quantum theory, changing the length $n$ leaves the rotation untouched: the same element simply reappears at spin $n/2$. And at fixed $n$, changing the rotation axis simply moves between the various classical and quantum theories.

Source: arXiv:2607.27173v1 - http://arxiv.org/abs/2607.27173v1 PDF: https://arxiv.org/pdf/2607.27173v1 Original Link: http://arxiv.org/abs/2607.27173v1