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This research project on optimal compression algorithms was concerned with several problems in the area of digital communications and the elimination of redundancy. The importance of the problem arises from the growing use of digital transmission in the military as well as the civilian communications field. There are many advantages enjoyed by digital transmission, the most notable being that there is almost no signal to noise degradation when relayed through a number of repeaters, whereas analog repeaters lose 3 db every time the number of repeaters is doubled. The one marked disadvantage of digital transmissions is that digitization of a basically analog source such as speech or TV results in a bandwidth expansion using conventional techniques. Good quality digitized voice requires on the order of 50,000 bits per second if data compression is not used. With conventional techniques, this requires 30 kHz of bandwidth while the analog voice signal could be transmitted over a 3 kHz channel. The area of bandwidth compression seeks to remove this disadvantage by eliminating the redundancy in the signal.