Square near triple arrays on twice as many symbols

Determine for which integers r ≥ 4 an (r × r, 2r)-near triple array exists.

Background

The question is motivated by unresolved entries in the paper’s enumeration table for near triple arrays. The parameter set has equal numbers of rows and columns and twice as many symbols as rows, and the authors identify it as a specific unresolved subfamily rather than resolving existence in general.

A solution would determine exactly which square instances in this family admit the prescribed near-equireplicate and near-intersection properties.

References

A more specialized question is motivated by the unresolved entries in Table \ref{tbl:nta=nbg}. For which $r\geq 4$ do $(r \times r, 2r)$-near triple arrays exist?

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

For which $r\geq 4$ do $(r \times r, 2r)$-near triple arrays exist?

— Near Triple Arrays  (2503.07166 - Gordeev et al., 10 Mar 2025) in Section “Concluding remarks,” Question environment beginning “For which r≥4”