最优非二进制单轨道格雷码
Optimal Non-Binary Single-Track Gray Code
AI总结：
研究非二进制单轨道格雷码，证明在有限域F_p（p = 3和p = 5）上存在特定长度和码字数量的此类码，对于更大素数p及非素数字母表大小m也有存在此类码的迹象。
AI中文摘要：
单轨道格雷码是一种循环格雷码，其码字长度为n，字母表大小为m，码字n个不同坐标对应的所有n个轨道都是第一个轨道的循环移位。在某些量化和编码应用中，此类码比传统格雷码更具优势。除非n = 2，不存在包含所有长度为n的2^n个码字的二进制码。本文证明，对于素数p = 3和p = 5，在有限域F_p上存在长度为p^t（n≥2）、具有p^p^t个码字的此类码。对于更大的素数p，t = 2时的合适码意味着对于任何t>2都存在此类码。若字母表大小m不是素数，也有迹象表明此类码存在。
英文摘要：
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.
