Balancing Cyclic R-ary Gray Codes II

New cyclic n-digit Gray codes are constructed over {0, 1, . . . , R − 1} for all R ≥ 2, n ≥ 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 ≥ 3 and n ≥ 2, every transition count is within 2 of the average R/n. For even R > 2, the codes are as close to uniform as possible, except when there are two anomalous transition counts for R ≡ 2 (mod 4) and R is divisible by n.