Existence classification for near balanced grids

Determine precisely for which triples of integers $(r,c,v)$ near balanced grids exist.

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 between near balanced grids and near triple arrays through an inequality involving intersection and covering-number sums.

Computational evidence suggests that near balanced grids are impossible for the majority of small parameter sets, but the paper does not provide a complete existence classification.

References

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

— Near Triple Arrays  (2503.07166 - Gordeev et al., 10 Mar 2025) in Question in Section 7, Concluding remarks (Section \ref{sec:concl})