Optimal quantizers for probability distributions on Sierpi\'nski carpets

In this paper, we investigate the optimal sets of $n$-means and the $n$th quantization error for singular continuous probability measures on $\mathbb R^2$ supported by Sierpi\'nski carpets. Utilizing the geometric structure of Sierpi\'nski carpets, first we determine the optimal sets of $n$-means for Borel probability measures $P$ on $\mathbb R^2$ supported by Sierpi\'nski carpets in progressive steps. Then, using suitable Voronoi partitions associated, we obtain the number of optimal sets of $n$-means as well as the $n$th quantization error involved in each case.