methods, unable to asymptotically guarantee that a block of data can be communicated free of error. Advanced techniques such as Reed–Solomon codes and, more recently, low-density parity-check (LDPC) codes and turbo codes, come much closer to reaching the theoretical Shannon limit, but at a cost of high computational complexity. Using these highly efficient codes and with the computing power in today's digital signal processors, it is now possible to reach very close to the Shannon limit. In fact, it was shown that LDPC codes can reach within 0.0045 dB of the Shannon limit (for binary additive white Gaussian noise (AWGN) channels, with very long block lengths). Mathematical statement The basic mathematical model for a communication system is the following: A message W is transmitted through a noisy channel by using encoding and decoding functions. An encoder maps W into a pre-defined sequence of channel symbols of length n. In its most basic model, the channel distorts each of these symbols independently of the others. The output of the channel –the received sequence– is fed into a decoder which maps the sequence into an estimate of the message. In this setting, the probability of error is defined as: Theorem (Shannon, 1948): 1. For every discrete memoryless channel, the channel capacity, defined in terms of the mutual information as has the following property. For any and , for large enough , there exists a code of length and rate and a decoding algorithm, such that the maximal probability of block error is . 2. If a probability of bit error is acceptable, rates up to are achievable, where and is the binary entropy function 3. For any , rates greater than are not achievable. (MacKay (2003), p. 162; cf Gallager (1968), ch.5; Cover and Thomas (1991), p. 198; Shannon (1948) thm. 11) Outline of proof As with the several other major results in information theory, the proof of the noisy channel coding theorem includes an achievability result and a matching converse result. These two components serve to bound, in this case, the set of possible rates at which one can communicate over a noisy channel, and matching serves to show that these bounds are tight bounds. The following outlines are only one set of many different styles available for study in information theory texts. Achievability for discrete memoryless channels This particular proof of achievability follows the style of proofs that make use