Enumeration and Generation of Cyclically Permutable Codes From Cyclic Codes

Cyclically permutable codes (CPCs) have found important applications in many communication systems, such as the multiple access collision channel without feedback, frequency-hopping spread spectrum communication channels and the digital watermarking systems. In this paper, by introducing a new method we completely settle the problem of constructing a CPC with the largest possible code size derived from a given simple-root cyclic code. The contribution of this paper is twofold. First, we present a new enumerative formula for the code size of such CPC with all the terms being positive integers, contrasting to the previously known ones given in Allard et al. (1973), Song et al. (1993), Tavares et al. (1971), and Xia and Fu (1997) which involve the Möbius function. Second, we provide an algebraic and systematic method to produce such a CPC. Several examples are also included to illustrate our main results.