Classical and Quantum MacWilliams Transforms as Spin Kinematics
2026-07-29 • Information Theory
Information Theory
AI summaryⓘ
The authors explain that the concept of spin plays a key role in understanding MacWilliams transforms, which are mathematical tools used in coding theory for both classical and quantum codes. They show that these transforms can be seen as a type of rotation (called a Wigner-D rotation) between two sets of basic elements. Importantly, this rotational view stays consistent even when the length of the code changes, and adjusting the rotation axis connects different coding theories. This perspective links classical and quantum coding through the language of spin and rotations.
SpinMacWilliams transformWeight enumeratorCoding theoryQubitsQuditsClassical codesWigner-D rotationError splittingRotation axis
Authors
Eric Kubischta, Ian Teixeira
Abstract
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.