Existence classification for near triple arrays

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

Background

The paper introduces near triple arrays as binary row-column designs in which symbol replication numbers and row-column, row-row, and column-column intersection sizes each take at most two consecutive values. It establishes both existence and non-existence results, including complete existence for all (3×c,v)(3\times c,v) with c≥6c\geq 6 and v≥cv\geq c, but notes that the general parameter classification remains incomplete.

The unresolved problem asks for a complete characterization across all admissible numbers of rows, columns, and symbols, extending the paper’s partial constructions, enumerations, and non-existence criteria.

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. \begin{question} For which combinations of $r, c, v$ do near triple arrays 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})