that all valid codewords produce a zero syndrome; teristics of the channel (particularly the random and burst error the syndrome then plays a crucial role in error correction. For the probabilities). The other important factors are likely to be the value systematic code given above, the optimum form of parity check of n/k (the increase in transmission rate over the channel), and the matrix is simply as in Equation 14.5. implementation complexity and cost of the coder and decoder. (14.5) 14.3 Block coding For unsystematic codes, construction of the parity check matrix is 14.3.1 Single parity checks more difficult. The simplest block coder appends a single parity digit to each block 14.3.3 Distance and code performance of k message digits. This produces a systematic (k + 1, k) code known as a single parity code. For binary digits, the parity bit may The Hamming distance between two codewords is simply the num- be either the modulo-2 sum of the message bits or 1 minus that sum. ber of bit positions in which they differ. If the Hamming distance The former case is known as even parity, because the sum of the between two codewords C1 and C2 is d, and C1 is transmitted, then d k + 1 bits of the codeword (including the parity bit) is 0, and the errors would have to occur for codeword C2 to be received. More latter is known as odd parity. generally, if the minimum Hamming distance between codeword C1 The decoder forms the sum of the bits of the received codeword. and any other valid codeword is dM and C1 is transmitted, then the For even parity a sum of 1 means that there has been an error, and a received codeword will not be a valid codeword if between 1 and sum of 0 is assumed to mean that the received codeword is correct. dm 1 errors occur. The decoder could therefore detect up to dM - (For odd parity 0 indicates an error, and the correct sum is 1). Note 1 errors. that when an error is detected there is no way to deduce which bit(s) Assume that an invalid codeword is received. The distance (num- are in error; also, if 