Existence of near balanced grids

Determine for which combinations of r, c, and v an (r × c, v)-near balanced grid exists.

Background

Near balanced grids are binary near-equireplicate row-column designs in which every pair of symbols occurs together in either of two consecutive numbers of rows and columns. The paper develops a duality relating near balanced grids to near triple arrays through an inequality involving aggregate intersection statistics.

Computational evidence suggests that near balanced grids are absent for the majority of tested parameter sets, but the authors do not provide a complete characterization of the parameters for which they exist.

References

It would be interesting to sharpen these results to a complete resolution of the following question. For which combinations of $r, c, v$ do near balanced grids exist?

— Near Triple Arrays  (2503.07166 - Gordeev et al., 10 Mar 2025) in Section 7, Concluding remarks, following the discussion of Theorem 5.1