Exact maximum periods of k-ary orientable sequences of orders 4 and 5

An orientable sequence has distinct cyclic windows even when the reading direction is unknown. We determine the maximum period of k-ary orientable sequences of order 4 for every even k ≥ 4, and of order 5 for every k ≥ 2. The formulas are (k⁴ − 2k² − 3k)/2 at order 4, and (k⁵ − 2k³ − k² + 2k)/2 or (k⁵ − 2k³ + k² − 4k)/2 at order 5 according as k is odd or even. Integer dual potentials and odd-cut inequalities give the upper bounds. Explicit rules on word equality patterns attain them; symbolic balance tables and connectivity arguments establish existence for all alphabet sizes in the stated domains. Further exact values are obtained at orders 6, 7 and 8. Fifteen entries of the comparison table of Mitchell and Wild are closed. The accompanying data include complete sequences and independent exact-arithmetic checkers. Contact: [email protected]. Article and data: CC BY 4.0; code: MIT License.