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Cyclically permutable codes have error-correcting properties which are invariant under arbitrary cyclic permutation of any of their code words. This paper summarizes the results of an empirical investigation of certain of these codes, which have parameters not covered by a previous paper of E. N. Gilbert. These codes are thought to be nearly optimal. Estimates of the obtainable number of code words are given. The codes may be suitable for use in certain asynchronous multiplex communication systems.