  We introduce finite-function-encoding (FFE) states which encode arbitrary
$d$-valued logic functions, i.e., multivariate functions over the ring of
integers modulo $d$, and investigate some of their structural properties. We
also point out some differences between polynomial and non-polynomial function
encoding states: The former can be associated to graphical objects, that we dub
tensor-edge hypergraphs (TEH), which are a generalization of hypergraphs with a
tensor attached to each hyperedge encoding the coefficients of the different
monomials. To complete the framework, we also introduce a notion of
finite-function-encoding Pauli (FP) operators, which correspond to elements of
what is known as the generalized symmetric group in mathematics. First, using
this machinery, we study the stabilizer group associated to FFE states and
observe how qudit hypergraph states introduced in arXiv:1612.06418v2 admit
stabilizers of a particularly simpler form. Afterwards, we investigate the
classification of FFE states under local unitaries (LU), and, after showing the
complexity of this problem, we focus on the case of bipartite states and
especially on the classification under local FP operations (LFP). We find all
LU and LFP classes for two qutrits and two ququarts and study several other
special classes, pointing out the relation between maximally entangled FFE
states and complex Butson-type Hadamard matrices. Our investigation showcases
also the relation between the properties of FFE states, especially their LU
classification, and the theory of finite rings over the integers.
