An improved upper bound on the Ruzsa number

Let $R_m$ be the least positive integer $r$ such that there exists a set $A\subseteq {\mathbb{Z}}_m$ with $A+A={\mathbb{Z}}_m$ for which the number of ordered solutions of $n=x+y$ with $x,y\in A$ is at most $r$ for every $n\in {\mathbb{Z}}_m$. In this note we prove that $R_m⩽128$ for every positive integer $m$, improving the previous bound $R_m⩽192$.