symmetric channel for simplicity) the most state at one time-step to another state at the next time step are likely coder path is the one which has the smallest number of determined by the shift register arrangement. The actual output code disagreements with the received bit stream. In a soft-decision digit stream is determined by both the shift register arrangement and decoder the most likely path is the one with the smallest squared the particular summations chosen to form the output digits. difference from the received signal. These measures of discrepancy There are M k patterns of k input digits, so from each state there between the received signal and possible transmitted signals are are M k possible next states. The behaviour of the coder can therefore referred to as path metrics. Encryption 14/11 output. The path stores within the decoder can therefore be im- plemented as 8 bit shift registers. State Time step 14.6.3 Code performance S1S2S3 A decoding error occurs when the path through the trellis selected by the decoder is not the correct path. The result will almost always be a burst error; it is even possible, for some codes, for the worst error to be infinite in extent. Such codes are clearly unusable. Analysis of the exact statistics of the error bursts (their probability as a function of their duration) is very complex; approximations can however be derived using the minimum free distance of the code. This is obtained by considering all possible error paths, that is, incorrect paths which depart from the correct path at one symbol time and rejoin it later. The minimum free distance is the minimum number of symbol differences between any of these error paths and the correct path. Note, however, that calculation of the true error rates involves analysis of many other interacting effects. In practice, the user of a convolutional code will normally use the published graphs of output BER as a function of SNR (E/. The coding gain is also often quoted, but it is important to remember that this varies with S