ancy is added. For An alternative to minimizing the probability of a word this, we use a channel code, which is a block code C of decoding error is to minimize the information symbol length n over Ax. We also use a channel encoding scheme, error rate Pe defined as Pe where which is an injective function from Ak to C. Using the p(i) = Pr{i#ui}. When the symbols are bits, the channel encoding scheme, blocks (u1, Uk) of source information symbol error rate is called the information symbols of length k are mapped to channel codewords of bit error rate (BER). Note that To mini- length N. The channel codewords are then sent over the mize the information symbol error rate, the decoder channel where errors may occur. Next, a function g from uses the symbol maximum a posteriori (MAP) rule, And to C called a channel decoding scheme, is used to map where for i = 1, k, the reconstructed information a received word y = (y1, ..., yn) to a channel codeword. symbol is computed as a symbol U E A that maximizes Finally, this channel codeword is mapped to the corre- the a posteriori probability Pr{U =u Y =y} Here U is sponding source word (), ) (since the encoding the random vector that corresponds to the information scheme is injective, this source word is unique when it symbol Ui, i = 1, k. exists). The rate of transmission of this system (or code The channel coding theorem [45] states that the rate of the code C) is k It characterizes the speed with source information can be transmitted reliably over which source information is transmitted over the channel a noisy channel, provided the rate of transmission is or equivalently the redundancy introduced by the below the capacity of the channel. In other words, any channel code. rate below the channel capacity is achievable. An ideal channel decoding scheme for this system Theorem 3.4.3 (Shannon's channel coding theorem). minimizes the probability of a word decoding error Consider a discrete memoryless channel with input al- phabet Ax and capacity C. For any positive number R<C and 