General formula for the dimension function of parameterized codes over arbitrary graphs

Determine a closed-form formula for the dimension dim_F C_X(d) of the parameterized evaluation code C_X(d) of order d associated with a projective toric subset X parameterized by an arbitrary simple graph G over a finite field F with q > 2; equivalently, determine the Hilbert function of the quotient ring S/I(X) in degree d for general G.

Background

The paper studies evaluation (parameterized) codes C_X(d) obtained by evaluating homogeneous polynomials of degree d on a projective toric subset X determined by a graph G. The dimension of C_X(d) equals the Hilbert function of S/I(X), where I(X) is the vanishing ideal of X.

While explicit formulas for the dimension exist in several special cases—when X is the projective torus (i.e., G is a forest or a non-bipartite unicyclic graph), for complete bipartite graphs, and for even cycles over fields with q = 3—there is no known general formula for arbitrary graphs. The authors highlight this gap explicitly as an unresolved issue.