A Note on Iterating an alpha-ary Gray Code

In this note we consider the number of distinct ff-ary codes produced by repeatedly applying the Gray code mapping of Sharma and Khanna [4]. This number was derived before by Lichtner [3], and we give a simpler proof here. Our key observation is a simple connection between this number and the period of binomial coefficients modulo ff. Then the result follows immediately from a known periodic property of binomial coefficients modulo ff [2, 5, 6].

A Note on Iterating an alpha-Ary Gray Code

. In this note we consider the number of distinct #-ary codes produced by repeatedly applying the Gray code mapping of Sharma and Khanna [Inform. Sci., 15 (1978), pp. 31--43]. This number was derived before by Lichtner [SIAM J. Discrete Math., 11 (1998), pp. 381--386], and we give an alternative proof here. Our key observation is a simple connection between this number and the period of binomial coe#cients modulo #. Then the result follows immediately from a known periodic property of binomial coe#cients modulo # [Fibonacci Quart., 27 (1989), pp. 64--79; SIAM J. Discrete Math., 9 (1996), pp. 55--62; Ann. Univ. Mariae Curie-Sklodowska Sect. A, 10 (1956), pp. 37--47]. Key words. gray code, binomial coe#cient AMS subject classifications. 68Q25, 68R01 PII. S0895480100367688 1