  In this note, we investigate the possibility of defining meaningful upper and
lower quantization dimensions for a compactly supported Borel probability
measure of order $r$, also for negative values of $r$. To this end, we utilize
the concept of partition functions, which generalizes the idea of the
$L^{q}$-spectrum and, in this way, naturally extends the work in [M.
Kesseb\"ohmer, A. Niemann, and S. Zhu. Quantization dimensions of probability
measures via R\'enyi dimensions. $\textit{Trans. Amer. Math. Soc.}$ 376.7
(2023), 4661$\unicode{x2013}$4678]. In particular, we provide natural fractal
geometric bounds as well as easily verifiable necessary conditions for the
existence of the quantization dimensions. Exact asymptotics for the absolutely
continuous case are also given.
