Optimal quantization for the one–dimensional uniform distribution with Rényi-$\alpha$-entropy constraints

summary:We establish the optimal quantization problem for probabilities under constrained Rényi-$\alpha$-entropy of the quantizers. We determine the optimal quantizers and the optimal quantization error of one-dimensional uniform distributions including the known special cases $\alpha = 0$ (restricted codebook size) and $\alpha = 1$ (restricted Shannon entropy)