n speed. This situation is changing with the increase of the speed of C-MOS elec- tronics and it is possible that soft decision decoders will also become available at optical speeds quite soon [60]. Due to the better exploitation of the information contained in the received message, gen- erally, soft decision performs better than hard decision, with a gain of the order of a couple of dBs at the typical optical SNo level. A hard decision-based decision device for a multilevel modulation format is based on the division of the space in which the multilevel constellation is created (e.g., the quadra- ture space or the Stokes space) in decision zones, let us call them D1 with j = 1, 2, M, each of which contains only one constellation point. Next Generation Transmission Systems The decision device simply associates the received signal to the constellation point inside the same decision zone. Naturally, to achieve good performances, the decision zones have to be optimized in order to minimize the overall error probability. Assuming that the noise distribution is isotropic around any constellation point, the opti- mum decision zones coincides with the so-called Voronoi zones of the multilevel constella- tion and they can be individuated with the so-called elementary cell method [46,61]. The method works as follows. In order to determine the Voronoi zone of the jth point of the constellation, as a first step the considered point is connected with M - 1 oriented segments with all the other constellation points. Let us call II the jth point and the segment connecting the jth point with the kth point. From now on, as in the notation , we will indicate with upper-line capital letters like AB vectors in an abstract space individuated by the end points (A and B in the exam- ple), coherently with the notation that we have adopted, where X indicates a vector in an abstract space, while E a physical vector (for example the electrical field). Calling Z the vector representing the received field, let us construct the projections of 