t digits for every k digits input, and the (1x1) = 1. code is described as 'a rate k/n code'. If the input digits are included unmodified in the coder output the Clearly, addition in this system is equivalent to the logical exclu- code is described as systematic. The additional digits introduced by sive OR (XOR) function, and multiplication is equivalent to the the coder are then known as parity or check digits. As well as the logical AND function. For non-binary systems, multiplication and conceptual attractiveness of systematic codes, they have the advan- addition operations can be implemented using either dedicated logic tage that a range of decoder complexities is made possible. The or lookup tables. 14/4 Coding Note that in modulo-2 arithmetic, addition and subtraction are errors is even, the single parity code will fail to detect them. This equivalent. Some textbooks only discuss binary coders, and there- code is therefore a single error detecting (SED) code. fore treat all subtractions as additions. However for values of M other than 2 addition and subtraction are not equivalent, and it is 14.3.2 Linear block codes essential to implement subtractions correctly. The even parity code is the simplest example of a powerful class of 14.2.4 Types of error codes called linear block codes. For such codes, the block of k message digits is represented as the k-element row vector d and the If the physical cause of digit errors is such that any digit is as likely n digit codeword produced by the coder is represented by the to be affected as any other, the errors are described as random. A n-element vector c. The function of the linear block coder is de- typical cause of random errors is thermal noise in the received scribed the Equation 14.1 where G is the (kxn) generator matrix. signal. Other types of interference, however, may make it likely that when an error occurs, several symbols in succession will be cor- (14.1) rupted; this is known as a burst error. A typical cause of burst errors is interference. Although th