Existence classification of near balanced grids

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

Background

Near balanced grids require the number of rows and columns containing each pair of symbols to take one of two consecutive values. The paper proves a duality relating near balanced grids and near triple arrays through a numerical comparison, and its computations suggest that near balanced grids are impossible for most small parameter sets. The authors explicitly leave a complete existence classification unresolved.

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 Question in Section 7, “Concluding remarks” (Section \ref{sec:concl})