SOME $\mathbb{Z}_{n-1}$ TERRACES FROM $\mathbb{Z}_{n}$ POWER-SEQUENCES, $n$ BEING AN ODD PRIME POWER

Abstract

A terrace for $\mathbb{Z}_m$ is a particular type of sequence formed from the $m$ elements of $\mathbb{Z}_m$. For $m$ odd, many procedures are available for constructing power-sequence terraces for $\mathbb{Z}_m$; each terrace of this sort may be partitioned into segments, of which one contains merely the zero element of $\mathbb{Z}_m$, whereas every other segment is either a sequence of successive powers of an element of $\mathbb{Z}_m$ or such a sequence multiplied throughout by a constant. We now refine this idea to show that, for $m=n-1$, where $n$ is an odd prime power, there are many ways in which power-sequences in $\mathbb{Z}_n$ can be used to arrange the elements of $\mathbb{Z}_n\setminus\{0\}$ in a sequence of distinct entries $i$, $1\le i\le m$, usually in two or more segments, which becomes a terrace for $\mathbb{Z}_m$ when interpreted modulo $m$ instead of modulo $n$. Our constructions provide terraces for $\mathbb{Z}_{n-1}$ for all prime powers $n$ satisfying $0\ltn\lt300$ except for $n=125$, $127$ and $257$.