Bounds for generalized triple-array width

Bound the quantity ω(r × c, v), defined as the least common upper bound on the three intersection-range widths of an (r × c, v)-generalized triple array, in terms of r, c, and v.

Background

Generalized triple arrays relax the near triple-array requirement by allowing each of the row-column, row-row, and column-column intersection sizes to lie in an interval of more than two consecutive integers. The quantity ω(r × c, v) measures the smallest uniform width sufficient for some generalized triple array with the given parameters.

The paper provides examples where relaxing one intersection width from 2 to 3 yields a design, but it does not establish general bounds for the minimum width.

References

For those parameter sets where near triple arrays do not exist, we investigated generalized triple arrays. In relation to this, we ask the following question. Let $\omega(r \times c, v)$ be the smallest integer $\omega$ such that there exists an $(r \times c, v; \omega_{rc}, \omega_{rr}, \omega_{cc})$-generalized triple array with $\omega_{rc}, \omega_{rr}, \omega_{cc} \leq \omega$. How can $\omega(r \times c, v)$ be bounded in terms of $r, c, v$?

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

In relation to this, we ask the following question. \begin{question} Let $\omega(r \times c, v)$ be the smallest integer $\omega$ such that there exists an $(r \times c, v; \omega_{rc}, \omega_{rr}, \omega_{cc})$-generalized triple array with $\omega_{rc}, \omega_{rr}, \omega_{cc} \leq \omega$. How can $\omega(r \times c, v)$ be bounded in terms of $r, c, v$? \end{question}

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

How can $\omega(r \times c, v)$ be bounded in terms of $r, c, v$?

— Near Triple Arrays  (2503.07166 - Gordeev et al., 10 Mar 2025) in Section “Concluding remarks,” Question environment beginning “Let ω(r × c, v)”