at nearby points corresponds to nearby bit words. In the case of squared constella- tions, tagging is easy due to the fact that the constellation can be divided into the Cartesian product of one dimensional constellations. If the number of one dimensional constellations is n (e.g., two for M-QAM, or four for M-4QAM), also the bit word has to be divided into n parts and each part is the tag of one one dimensional constellation. This tagging technique is shown in Figure 9.37 in the case of 16-QAM. In the case of more complex configurations, tagging can be much more complicated, and it is also pos- sible that in a few cases the adjacency of symbols corresponding to very different bit words cannot be avoided. An upper bound for the number of erroneous bits in a wrong symbol can be obtained assuming that, when a symbol is wrong, the receiver produces a random symbol among the M - 1 different from the transmitted one. In this case, if n = logM, Mlog2M (9.48) 2(M-1) Telecommunication Networks FIGURE 9.37 Tagging of a squared constellation. where besides the definition of n the following combinatorial property has been used: 2(i)-2 (9.49) Thus, the following upper bound can be used for the error probability, if the symbol error probability is known: (9.50) As far as the symbol error probability is concerned, it depends on the decision criterion adopted to produce an estimate of the transmitted symbol. We will consider here only the so-called "hard decision" criteria, where the estimate of the transmitted symbol is produced by the decision device without taking into account the presence of a FEC. After hard decision, the produced symbol stream (or often the produced bit stream after multilevel decoding) is passed to the FEC for error correction [43,60]. Traditionally, soft decision criteria, where decision and FEC decoding are not divided, but the decision device uses the FEC redundancy also to take the better possible estimate of the transmitted message, has been considered too complex to be used at optical trans- missio