Explicit characterization of reductions for general monomial–Cartesian codes

Determine, for general monomial–Cartesian codes defined by evaluation sets Z = Z_1 \times \cdots \times Z_m and the ideal I generated by Q_j(X_j) = \prod_{\beta \in Z_j}(X_j - \beta), an explicit characterization or algorithm for the support of the reduced polynomial h equivalent to X^{\mathbf e} modulo I when \mathbf e \notin E. Specifically, ascertain the structure of \operatorname{supp}(h) in the quotient ring R = \mathbb{F}_q[X_1,\dots,X_m]/I to make the reduction of Minkowski sums explicit beyond known special cases.

Background

A central step in relating Schur products of evaluation codes to Minkowski sums of exponent sets is reducing monomial products modulo the defining ideal. In special cases (toric codes and Reed–Muller/hyperbolic codes), explicit reductions are known, enabling exact Schur product descriptions.

For general monomial–Cartesian codes, the authors state that the support of the reduced representative h equivalent to X^{\mathbf e} modulo I is not known. This gap motivates their focus on J-affine variety codes, for which they provide explicit reductions.