Optimal Non-Binary Single-Track Gray Code

A single-track Gray code is a cyclic Gray code with codewords of length $n$, over an alphabet of size $m$, such that all the $n$ tracks that correspond to the $n$ distinct coordinates of the codewords are cyclic shifts of the first track. Such codes have advantages over the conventional Gray codes in certain quantization and coding applications. Unless $n=2$, there are no such binary codes that contain all the $2^n$ codewords of length $n$. In this paper, we prove that such codes of length $p^t$, $n \geq 2$, with $p^{p^t}$ codewords, over $\F_p$, $p$ prime, exist, for $p=3$ and $p=5$. For larger prime $p$ an appropriate code for $t=2$, implies the existence of such a code for any $t >2$. If the alphabet size $m$ is not a prime there are also indications that such codes exist.