In this paper, we build upon the analysis initiated by Gennadiy Averkov \& Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$, thereby improving the earlier bound of $O(m^4 Δ)$. Recall that a matrix is called $Δ$-modular if the maximum absolute value of every $m \times m$ minor is exactly $Δ$.

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