In this paper, we build on the analysis initiated by Averkov and Schymura (2022) and prove that the shifted Heller constant for matrices with $m$ rows is at most $m^2 + m$ for all sufficiently large $m$. Consequently, the number of distinct columns of a $Δ$-modular matrix $A \in Z^{m \times n}$ of rank $m$ is at most $O(m^2 Δ)$. This improves upon the earlier bounds of $O(m^4 Δ)$ and $O(m^2 Δ^2)$.

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