For the mean-square error distortion criterion, it can be easily shown that the optimal set of reconstruction values formula_93 is given by setting the reconstruction value formula_13 within each interval formula_49 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 formula_97, 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 formula_100. In some designs, rather than optimizing for a particular number of classification regions formula_40, the quantizer design problem may include optimization of the value of formula_40 as well. For some probabilistic source models, the best performance may be achieved when formula_40 approaches infinity. In the above formulation, if the bit rate constraint is neglected by setting formula_84 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 formula_63 alone. The indices produced by an formula_40-level quantizer can be coded using a fixed-length code using formula_107 bits/symbol. For example, when formula_108256 levels, the FLC bit rate formula_62 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. Assuming an FLC with formula_40 levels, the Rate–Distortion minimization problem can be reduced to distortion minimization alone. The reduced problem can be stated as follows: given a source formula_65 with pdf formula_66 and the constraint that the quantizer must use only formula_40 classification regions, find the decision boundaries formula_114 and reconstruction levels formula_115 to minimize the resulting distortion   Finding an optimal solution to the above problem results in a quantizer sometimes called a MMSQE (minimum mean-square quantization error) solution, and the resulting pdf-optimized (non-uniform) quantizer is referred to as a "Lloyd–Max" quantizer, named after two people who independently developed iterative methods to solve the two sets of simultaneous equations resulting from formula_117 and formula_118, as follows: which places each threshold at the midpoint between each pair of reconstruction values, and which places each reconstruction value at the centroid (conditional expected value) of its associated classification interval.