What is d-dimensional digit-shift Weyl gate? Does this gate has other names?

Question

I was reading this paper: Quantum Anonymous Networking: A Quantum Leap in Privacy (DOI: 10.1109/MNET.2024.3408814)

In 1) Anonymous Data Disseminating and 3) Anonymous Qudit Teleporting section of the paper, it mentions (d-dimensional) digit-shift Weyl gate:

In 1) Anonymous Data Disseminating section:

and then sequentially performing controlled d-dimensional Weyl-x, i.e., digit-shift $ X_{d}^{j} $ gates with the first qudit as the control and each of the subsequent qudits as the target

The paper further explains on this part by saying:

In this controlled digit-shift operation, if the control qudit is in the state |j〉, the target qudit is shifted by j in the computational basis applying j times the d-dimensional digit-shift Weyl gate $ X_d $

In 3) Anonymous Qudit Teleporting section:

Alice applies the controlled digit-shift Weyl gate, i.e., the d-dimensional controlled-NOT (CNOT) gate between her teleporting qudit (control) | ψ 〉 T and her member qudit (target)

Given that the paper use "i.e.", so I think that: controlled d-dimensional Weyl-x = controlled digit-shift $ X_{d}^{j} $ gates = controlled (d-dimensional) digit-shift Weyl gate = d-dimensional controlled-NOT (CNOT) gate

But I still have no idea what these gates are.

I had tried searching with these terms:

• digit shift quantum gate
  • Only get results about quantum gate for qubit
    • The first 2 result is wikipedia. Quantum logic gate & List of quantum logic gates
    • List of quantum logic gates has phase shift, but it doesn't contain digit shift.
• digit shift quantum
  • How to realize the index shift operation in quantum circuit? - quantumcomputing stackexchange
    • This question is about index shift not digit shift. Is "index shift" the same as "digit shift"? I don't know.
  • Quantum Shift Register
    • It works on qubit not qudit.
    • I can't understand it though ...
  • Quantum circuit implementation of shift operator in quantum walk
    • Seems releated, but I can't understand it.
• Weyl-x gate
  • Quantum Gates pdf by Gavin E. Crooks
    • Excellent resource for me to understand quantum computing.
    • But still talk about qubit not qudit.
    • Only contains info about Weyl chamber, not weyl-x gate

None of these results answer my question.

Could someone explain what is d-dimensional digit-shift Weyl gate and its operation?