Optimum block codes for noiseless input restricted channels

Abstract

A method for determining maximum-size block codes, with the property that no concatenation of codewords violates the input restrictions of a given channel, is presented. The class of channels considered is essentially that of Shannon (1948) in which input restrictions are represented through use of a finite-state machine. The principal results apply to channels of finite memory and codes of length greater than the channel memory but shorter codes and non-finite memory channels are discussed briefly. A partial ordering is first defined over the set of states. On the basis of this ordering, complete terminal sets of states are determined. Use is then made of Mason's general gain formula to obtain a generating function for the size of the code which is associated with each complete terminal set. Comparison of coefficients for a particular block length establishes an optimum terminal set and codewords of the maximum-size code are then obtained directly. Two important classes of binary channels are then considered. In the first class, an upper bound is placed on the separation of 1's during transmission while, in the second class, a lower bound is placed on this separation. Universal solutions are obtained for both classes.