Assuming an FLC with levels, the rate–distortion minimization problem can be reduced to distortion minimization alone. The reduced problem can be stated as follows: given a source with PDF and the constraint that the quantizer must use only classification regions, find the decision boundaries and reconstruction levels 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 and , 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. Lloyd's Method I algorithm, originally described in 1957, can be generalized in a straightforward way for application to vector data. This generalization results in the Linde–Buzo–Gray (LBG) or k-means classifier optimization methods. Moreover, the technique can be further generalized in a straightforward way to also include an entropy constraint for vector data. Uniform quantization and the 6 dB/bit approximation The Lloyd–Max quantizer is actually a uniform quantizer when the input PDF is uniformly distributed over the range . However, for a source that does not have a uniform distribution, the minimum-distortion quantizer may not be a uniform quantizer. The analysis of a uniform quantizer applied to a uniformly distributed source can be summarized in what follows: A symmetric source X can be modelled with , for and 0 elsewhere. The step size and the signal to quantization noise ratio (SQNR) of the quantizer is . For a fixed-length code using bits, , resulting in , or approximately 6 dB per bit. For example, for =8 bits, =256 levels and SQNR = 8×6 = 48 dB; and for =16 bits, =65536 and SQNR = 16×6 = 96 dB. The property of 6 dB improvement in SQNR for each extra bit used in quantization is a well-known figure of merit. However, it must be used with care: this derivation is only for a uniform quantizer applied to a uniform source. For other source PDFs and other quantizer designs, the SQNR may be somewhat different from that predicted by 6 dB/bit, depending on the type of PDF, the type of source, the