Perfect ruler

A perfect ruler of length ${\displaystyle \ell }$ is a ruler with integer markings ${\displaystyle a_{1}=0<a_{2}<\dots <a_{n}=\ell }$, for which there exists an integer ${\displaystyle m}$ such that any positive integer ${\displaystyle k\leq m}$ is uniquely expressed as the difference ${\displaystyle k=a_{i}-a_{j}}$ for some ${\displaystyle i,j}$. This is referred to as an ${\displaystyle m}$-perfect ruler.

An optimal perfect ruler is one of the smallest length for fixed values of ${\displaystyle m}$ and ${\displaystyle n}$.