Replace ILP lower bounds for b_2(5,2,2;s) with theoretical proofs

Establish theoretical lower bounds that replace some of the integer linear programming computations used to prove the exact values of b_2(5,2,2;s), the minimum number of lines in PG(4,2) such that every plane contains at least s lines.

Background

The paper determines b_2(5,2,2;s) for all s by combining explicit constructions with lower bounds obtained partly through integer linear programming computations.

Although the numerical values are fully determined, the authors identify the replacement of some computational certificates by theoretical lower-bound arguments as an unresolved problem. They mention techniques from earlier work on blocking sets as a possible blueprint.

References

However, this result is still based on integer linear programming computations and we propose it as an open problem to replace some of these by theoretical lower bounds.

— Generalized Hamming weights of additive codes and geometric counterparts  (2512.16327 - D'haeseleer et al., 18 Dec 2025) in Section 8, Conclusion and open problems