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Hamada conjecture on minimum p-rank of designs

Prove that, for any prime p and any 2-(m, s, λ) design, the p-rank of the incidence matrix is minimized by the affine geometric designs (equivalently, by the duals of Reed–Muller codes), thereby establishing Hamada’s conjecture in full generality.

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Background

A 2-(m, s, λ) design is a collection of s-element blocks on an m-point set such that every pair of points appears in exactly λ blocks. The p-rank of a design is the rank over the finite field F_p of its incidence matrix.

Hamada (1973) conjectured that affine geometric designs (the duals of Reed–Muller codes) minimize the p-rank among all designs with the same parameters. While special cases have been resolved and this paper confirms the case s = 4 up to a constant factor in the exponent via design 3-LCC lower bounds, the full conjecture remains open.

References

In 1973, Hamada made a foundational conjecture (see for a recent survey) in the area that states that affine geometric designs (i.e., duals to the Reed--Muller LCCs) minimize the $p$-rank among all algebraic designs of the same parameters.

— Exponential Lower Bounds for Smooth 3-LCCs and Sharp Bounds for Designs (2404.06513 - Kothari et al., 9 Apr 2024) in Section 1, Introduction (Connections to the Hamada Conjecture)