Exact threshold for q-ary Deza-type systems

Determine the exact value of f_q(k,l), the smallest integer m such that every (n,k,l)_q-family of cardinality greater than m is a Δ_q(n,k,l)-system.

Background

The paper defines f_q(k,l) as the least cardinality threshold forcing an (n,k,l)_q-family to have the q-ary Δ-system structure. Its q-ary extension of Deza’s theorem establishes the upper bound f_q(k,l)≤4(k-l)^2+2(k-l)+1 under the condition that max{2l+2,q-1} is no larger than this quantity. The sharp threshold, including whether this bound can be improved and what the exact dependence on q, k, and l is, remains unresolved.