On One-to-One Codes for Memoryless Cost Channels

A new twist to the classical problem of compressing a discrete source over noiseless channel alphabets where different code symbols may have different transmission costs is considered. One-to-one or nonsingular codes map each letter from the source alphabet into a distinct codeword consisting of a string of channel alphabet letters; they need not be uniquely decodable. They have been studied for binary code alphabets with equal letter costs since the early 1970s and are extended here to memoryless cost channel alphabets. The study of optimal one-to-one codes is linked in this paper to three other well-known source coding problems. The algorithm to construct the optimal one-to-one code is closely related to the Varn code and the Tunstall code. These relationships and convexity arguments are used to establish some upper and lower bounds on the expected cost per source symbol of the optimal one-to-one code. Finally, the average cost per symbol of the optimal one-to-one code is demonstrated to offer a new lower bound to the average cost per symbol of the optimal uniquely decodable code which is sometimes tighter than the classic bound from Shannon's landmark 1948 paper.