Hamada’s p-rank minimization conjecture for designs

Establish that, for any prime p and any 2-(m,s,λ) design, the affine geometric designs (equivalently, duals of Reed–Muller codes with corresponding parameters) minimize the p-rank among all algebraic designs with the same parameters.

Background

The authors discuss a longstanding conjecture in algebraic design theory—Hamada’s conjecture—linking combinatorial designs and coding theory. A 2-(m,s,λ) design’s p-rank is the rank over F_p of its incidence matrix; the conjecture posits that affine geometric designs minimize this p-rank among all designs with the same parameters.

This conjecture is deeply connected to LCC lower bounds via a transformation that relates design incidence structures to locally correctable codes. The paper confirms the conjecture up to constants for s=4 (the 3-LCC regime) by proving sharp lower bounds for design 3-LCCs, but the general conjecture remains unresolved.