Beelen–Datta–Ghorpade projective generalized Hamming-weight conjecture

Determine the exact value of the projective generalized Hamming weight quantity e^P_r(d,m;q) for 1 ≤ r ≤ \binom{m+d}{d} and q ≥ d+1, by proving that e^P_r(d,m;q)=H_r(d,m;q)+\pi_{m-l-1}(q), where l is the smallest index with nonzero coordinate in the lexicographically r-th largest element w_r(d,m) of \Omega(d,m).

Background

The paper recalls the Beelen–Datta–Ghorpade conjecture as the proposed exact formula for the maximum number of finite-field rational points in the common projective zero set of an r-dimensional space of degree-d homogeneous polynomials in m+1 variables. The formula is known in several special cases, including r=1, r=2, d=2, m=1, m=2, and sufficiently large q, but is not established in full generality for q ≥ d+1.

This conjecture is the k=m−1 special case of the paper’s broader k-dimensional projective conjecture. The paper also proves that its proposed projective conjecture in dimension k=m−2 would imply the Beelen–Datta–Ghorpade conjecture.

References

Beelen, Datta and Ghorpade conjectured a new exact formula for $e_r{P}(d,m;q)$.

— A $k$-Dimensional Version of the Largest Intersection Problem  (2608.17771 - Lin, 18 Aug 2026) in Conjecture (Conj: complete GDC), Section 1

Then we have

= +\pi_{s}(q).

\end{conjecture}

— A $k$-Dimensional Version of the Largest Intersection Problem  (2608.17771 - Lin, 18 Aug 2026) in Conjecture (Projective conjecture), Section 1, subsection "k-dimensional conjecture"

The generalized Hamming weights $\mathrm{d}r\bigl(\mathcal{C}_d(X{\mathrm{Car}})\bigr)$ of $\mathcal{C}d(X{\mathrm{Car}})$ are known , and we conjecture that projective Cartesian codes have the smallest generalized Hamming weights among evaluation codes supported on reduced complete intersections with the same degrees.

— Generalized Hamming weights of codes arising from complete intersection  (2608.19978 - Moreno et al., 20 Aug 2026) in Section 3, “A more general conjecture,” Conjecture \ref{conj:ghws}