e true behaviour of the channel may be The multiplications and additions in this equation are carried out more complex than either of these simple models, the random error in GF(M) arithmetic (i.e. modulo-2 for binary digits). The code- and burst error models are simple, effective, and universally used words generated by the equation are called valid codewords. Since for describing channel characteristics and error control code per- there are 2k possible datawords, only 2k of the 2nd possible n-digit formance. words are valid codewords. A channel used for transmitting binary digits is known as a binary Systematic linear block codes are produced by a generator matrix channel, and if the probability of error is the same for Os and 1s, the of the form shown in Equation 14.2, where LX is the k) unit channel is called a Binary Symmetric Channel. The probability of matrix. error in binary digits is known as the bit error rate (BER). G = [IKIP] (14.2) 14.2.5 Coding gain When G has this form, the codeword c has the form of Equation Coding gain is a parameter commonly used for evaluating the 14.3, in other words the first k digits of the codeword equal the effectiveness of an error correcting code, and hence for comparing dataword, and the last n - k digits are parity digits. codes. It is defined as the saving in energy per source bit of information for the coded system, relative to an uncoded system c = [d|dP] (14.3) delivering the same BER. The effects of both error correction and the increase in transmission rate by a factor of n/k must be included Let I be the received codeword; in the absence of errors The when calculating the coding gain. decoder performs the operation of Equation 14.4, where H is the n) parity check matrix, to produce the k) element 14.2.6 Criteria for choosing a code syndrome S. The primary objective of error control coding will be to achieve a (14.4) desired end-to-end probability of either uncorrected or undetected digit errors. The choice of code will depend on the error charac- H is chosen so 