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{\bf Mary Flahive }
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{\bf Balancing Cyclic $R$-ary Gray Codes$\,$ II}
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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 $n$.





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