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Effective compression technology is crucial for 3DGS to adapt to varying storage and transmission conditions. However, existing methods fail to address size constraints while maintaining optimal quality. In this paper, we introduce SizeGS, a framework that compresses 3DGS within a specified size budget while optimizing visual quality. We start with a size estimator to establish a clear relationship between file size and hyperparameters. Leveraging this estimator, we incorporate mixed precision quantization (MPQ) into 3DGS attributes, structuring MPQ in two hierarchical level -- inter-attribute and intra-attribute -- to optimize visual quality under the size constraint. At the inter-attribute level, we assign bit-widths to each attribute channel by formulating the combinatorial optimization as a 0-1 integer linear program, which can be efficiently solved. At the intra-attribute level, we divide each attribute channel into blocks of vectors, quantizing each vector based on the optimal bit-width derived at the inter-attribute level. Dynamic programming determines block lengths. Using the size estimator and MPQ, we develop a calibrated algorithm to identify optimal hyperparameters in just 10 minutes, achieving a 1.69\(\times\) efficiency increase with quality comparable to state-of-the-art methods.