Iterating an $\alpha$-ary Gray code

• Authors: Jonathan Lichtner
• Venue: SIAM Journal on Discrete Mathematics
• Year: 1998

In this paper we prove a theorem on the number of distinct codes produced when the $\alpha$-ary Gray code mapping of Sharma and Khanna [Inform. Sci., 15 (1978), pp. 31--43] is iteratively applied to an $\alpha$-ary, dimension l code; that is, starting with an $\alpha$-ary, dimension l code, and repeatedly applying the permutation given by Sharma and Khanna's mapping. From this theorem, it is easy to show there are $\Theta(l^q)$ distinct codes generated from this mapping, where q is the number of distinct primes in $\alpha$ (Let $f:\mbox{\bf{N}} \rightarrow \mbox{\bf{R}}^{*}. O(f)$ is the set of functions $g:\mbox{\bf{N}} \rightarrow \mbox{\bf{R}}^{*}$ such that for some $c \in \mbox{\bf{R}}^{+}$ and some $n_0 \in \mbox{\bf{N}}, g(n) \leq cf(n)$ for all $n \geq n_0$. $\Theta(f)$ is the set of functions $g:\mbox{\bf{N}} \rightarrow \mbox{\bf{R}}^{*}$ such that g is in O(f) and f is in O(g).). To prove this theorem we show that any base $\alpha$, dimension l code word will cycle in O(lq) iterations of this Gray code mapping, and that this upper bound is attained. This theorem is a generalization of a theorem proven by Culberson [Evolutionary Comput., 2 (1995), pp. 279--311] for the binary case.