Existence of near triple arrays

Determine for which combinations of positive integers r, c, and v an (r × c, v)-near triple array exists.

Background

Near triple arrays are binary row-column designs with near-equireplicate symbol occurrences and pairwise row-column, row-row, and column-column intersection sizes restricted to two consecutive values. The paper establishes several existence and non-existence results, including complete existence for three-row arrays when c ≥ 6 and v ≥ c, but the authors state that the overall existence landscape remains incomplete.

The question asks for a full characterization of the admissible parameter triples (r,c,v), encompassing both constructive existence results and structural non-existence conditions.

References

In the present paper, we gave several partial results on the following question, both positive and negative, but the picture is far from complete. For which combinations of $r, c, v$ do near triple arrays exist?

— Near Triple Arrays  (2503.07166 - Gordeev et al., 10 Mar 2025) in Section 7, Concluding remarks, Question

For which combinations of $r, c, v$ do near triple arrays exist?

— Near Triple Arrays  (2503.07166 - Gordeev et al., 10 Mar 2025) in Section “Concluding remarks,” Question environment immediately preceding the conjecture