Complete parameterisation of dual-unitary two-qudit gates for q>2

Develop a complete explicit parameterisation of all two-qudit dual-unitary gates for local dimension q>2, extending the known exhaustive parameterisation for q=2.

Background

For q=2, a complete parameterisation of all two-qubit dual-unitary gates is known, and many constructions exist for q>2, including families based on direct sums, complex Hadamard matrices, and other algebraic structures.

Despite these families, an exhaustive and explicit parameterisation covering all dual-unitary gates for general q>2 has not been obtained.