 coding gains. In wireless-cellular applications, for instance, the two convolutional encoders are typically recursive, with a constraint length m = 4 and a code rate 1/3, yielding overheads better than 20%. To quote some representative figures, the coding gain of turbo codes at BER = 10-6 and after 10 iterations may be over 2 dB better than that of concatenated codes, 4.5 dB better than that of convolutional codes, and represent a 10 dB SNR improvement with respect to the uncoded BER case. As two illustrative examples, suffice it to mention here the significant applications of turbo codes in the fields of aerospace and satellite communications10 and 3G/cellular telephony (UMTS, cdma2000). 11 This completes our "high-level" overview of the various ECC types and families. Since ECCs can only correct error patterns with up to w = {(dmin - 1)/2} bit errors, it is clear that, in the general case, the bit-error rate (BER) is never identical to zero. Bit errors are generated when the receiver makes the "wrong decision," namely out- puts a 1 when the payload bit under consideration was a 0, and the reverse. Here, I shall not describe the different types of receiver architectures and decision techniques. Both are intimately dependent upon the type of signal modulation formats, the method of encoding bits into actual physical signal waveforms, whether electrical, radio, or optical. The ECC algorithms pervade all communication and information systems, and at practically all protocol layers: in point-to-point radio links (ground, aeronautical, satellite, and deep-space), in optical links (from fiber-to-the-home to metropolitan or core networks and undersea cable systems), in mobile cellular networks (GSM, GPRS, CDMA, UMTS/3G), in broadband wireless or wireline access (ADSL, 802.11/WiFi), in metropolitan, regional, and global voice or data transport (Ethernet, ATM, TCP/IP, or 9 www.aero.org/publications/crosslink/winter2002/04.html www.aero.org/publications/crosslink/winter2002/04.html, See, for instance: http://users.tkk.fi