Characterize additional minimum-weight codewords for two-maximal-monomial decreasing codes

Characterize when the minimum-weight codewords of a monomial decreasing evaluation code on the full grid F_q^2 are exactly those arising from the two individual decreasing rectangles, for decreasing supports with two maximal monomials of equal footprint value; equivalently, determine precisely when the associated reduced extremal-zero set contains no noncanonical extremal polynomial.

Background

For a decreasing monomial set with two maximal monomials of equal footprint value, the paper reduces the classification of extremal polynomials—and hence of minimum-weight codewords—to a smaller decreasing support whose maximal monomials are relatively prime, typically of the form {x{d_x}, y{d_y}}. Theorem \ref{thm.totalcount} decomposes the minimum-weight codewords into those coming from the two rectangular subcodes and those induced by noncanonical extremal triples in the reduced support.

The paper proves that the integrality condition (q-a_1)/d_x = (q-b_2)/d_y is necessary for additional minimum-weight codewords, but Example \ref{ex.11} shows that this condition is not sufficient. Thus, the unresolved task is to give a complete characterization of when the reduced extremal-zero set is empty, or equivalently when no additional minimum-weight codewords occur beyond the two rectangular families.

References

A complete characterization of when

\MW(\mathcal C(F_q2,\mathcal M))= \MW(\mathcal C(F_q2,\mathcal M_1))\cup \MW(\mathcal C(F_q2,\mathcal M_2))

is still open.

— Polynomials with restricted support and maximal zeros on a finite Cartesian set  (2610.05813 - Bhunia et al., 5 Oct 2026) in Remark following Corollary in Section "Applications to Monomial Decreasing Codes", subsection "Two maximal monomials"