Optimal single-track Gray codes from Gray-ordered necklaces

Overview & Synopsis

A single-track Gray code of length $n$ is a cyclic Gray code of $n$-bit words in which every column is a cyclic shift of the first. For prime $n$ the longest possible period is $2^n-2$. Whether it is attained for every prime has been open since Hiltgen, Paterson and Brandestini introduced these codes in 1996, and optimal codes were known for primes $n\le 19$. We show that a conjecture of Degni and Drisko (2007) on the reflected Gray code order of necklaces implies optimal single-track Gray codes for every prime. Their list of necklaces, conjecturally a Gray path for prime $n$, closes into a Hamilton cycle of the necklace graph after $(n-3)/2$ explicit Pósa rotations, and such a cycle lifts to an optimal code. This implication is formalized in Lean 4. Together with computation it gives optimal single-track Gray codes for every odd prime $n\le 47$: for $n\le 37$ explicit codes checked word by word by a program independent of the generator, and for $n=41$, $43$ and $47$ codes obtained from the implication after the Degni-Drisko property was verified at these lengths by two independently written programs.