Equivalence of extremal type-three matrix constructions

Determine whether the matrices \(M(a,b)\) of type \(3\) having \(\Delta+4\) columns and parameter \(a_3\) in the stated admissible range are inequivalent, in the sense of Definition 4.5 of the cited prior work, for distinct values of \(a_3\).

Background

The paper identifies a family of type-three matrices that can attain the upper bound $\Delta+4$ when $\Delta\equiv2\pmod 6$. These matrices are parameterized by values of $a_3$ satisfying $\frac23\Delta\leq a_3\leq\frac34\Delta+1$ and $a_3\equiv1\pmod3$.

It remains unresolved whether different parameter values produce genuinely inequivalent matrices under the equivalence relation used in the authors’ earlier work.