Balancing Cyclic $R$-ary Gray Codes$\,$ II

Abstract

New cyclic $n$-digit Gray codes are constructed over $\{0, 1, \ldots, R-1 \}$ for all $R \ge 2$, $n \ge 3$. These codes have the property that the distribution of digit changes (transition counts) between two successive elements is close to uniform. For $R=2$, the construction and proof are simpler than earlier balanced cyclic binary Gray codes. For $R \ge 3$ and $n \ge 2$, every transition count is within $2$ of the average $R^n/n$. For even $R >2$, the codes are as close to uniform as possible, except when there are two anomalous transition counts for $R \equiv 2 \pmod{4}$ and $R^n$ is divisible by $n$.

Combinatorial Gray Codes — an Updated Survey

“…Recently, Herter and Rote [HR18] devised loopless algorithms for generating b-ary Gray codes based on generalizations of the Tower of Hanoi puzzle. Flahive [Fla08] describes a construction of balanced b-ary Gray codes with transitions counts that differ by at most 2 from the average b n /n, improving upon the earlier paper by Flahive and Bose [FB07]. P20 Many other of the properties discussed before for the binary case (weight-monotonicity, runs lengths, trend-freeness, single-track etc.)…”

Reordering columns for smaller indexes

“…Non-recursive orders. There are balanced and nearly balanced Gray codes [21,22,37]. Unlike the other types of Gray codes, the number of runs in all columns is nearly the same when sorting all possible tuples for N 1 = N 2 = .…”