  In recent work [arXiv:2003.06939v2] a novel fermion to qubit mapping --
called the compact encoding -- was introduced which outperforms all previous
local mappings in both the qubit to mode ratio, and the locality of mapped
operators. There the encoding was demonstrated for square and hexagonal
lattices. Here we present an extension of that work by illustrating how to
apply the compact encoding to other regular lattices. We give constructions for
variants of the compact encoding on all regular tilings with maximum degree 4.
These constructions yield edge operators with Pauli weight at most 3 and use
fewer than 1.67 qubits per fermionic mode. Additionally we demonstrate how the
compact encoding may be applied to a cubic lattice, yielding edge operators
with Pauli weight no greater than 4 and using approximately 2.5 qubits per
mode. In order to properly analyse the compact encoding on these lattices a
more general group theoretic framework is required, which we elaborate upon in
this work. We expect this framework to find use in the design of fermion to
qubit mappings more generally.
