been published for only three PDFs: the uniform, exponential, and Laplacian distributions. Iterative optimization approaches can be used to find solutions in other cases. Note that the reconstruction values affect only the distortion – they do not affect the bit rate – and that each individual makes a separate contribution to the total distortion as shown below: where This observation can be used to ease the analysis – given the set of values, the value of each can be optimized separately to minimize its contribution to the distortion . For the mean-square error distortion criterion, it can be easily shown that the optimal set of reconstruction values is given by setting the reconstruction value within each interval to the conditional expected value (also referred to as the centroid) within the interval, as given by: . The use of sufficiently well-designed entropy coding techniques can result in the use of a bit rate that is close to the true information content of the indices , such that effectively and therefore . The use of this approximation can allow the entropy coding design problem to be separated from the design of the quantizer itself. Modern entropy coding techniques such as arithmetic coding can achieve bit rates that are very close to the true entropy of a source, given a set of known (or adaptively estimated) probabilities . In some designs, rather than optimizing for a particular number of classification regions , the quantizer design problem may include optimization of the value of as well. For some probabilistic source models, the best performance may be achieved when approaches infinity. Neglecting the entropy constraint: Lloyd–Max quantization In the above formulation, if the bit rate constraint is neglected by setting equal to 0, or equivalently if it is assumed that a fixed-length code (FLC) will be used to represent the quantized data instead of a variable-length code (or some other entropy coding technology such as arithmetic coding that is better than an FLC in the rate–distortion sense), the optimization problem reduces to minimization of distortion alone. The indices produced by an -level quantizer can be coded using a fixed-length code using bits/symbol. For example, when 256 levels, the FLC bit rate is 8 bits/symbol. For this reason, such a quantizer has sometimes been called an 8-bit quantizer. However using an FLC eliminates the compression improvement that can be obtained by use of better entropy coding.