Difference bases in cyclic and dihedral groups

A subset $B$ of a group $G$ is called a {\em difference-basis} of $G$ if each element $g\in G$ can be written as the difference $g=ab^{-1}$ of some elements $a,b\in B$. The smallest cardinality $|B|$ of a difference basis $B\subset G$ is called the {\em difference weight} of $G$ and is denoted by $\Delta[G]$. We prove that for every $n\ne 4$ the cyclic group $C_n$ has difference weight $\frac{1+\sqrt{4|C_n|-3}}2\le \Delta[C_n]\le\sqrt{2n}$ and for every $n\in\mathbb N$ the dihedral group $D_{2n}$ has difference weight $\sqrt{2|D_{2n}|}\le \Delta[D_{2n}]\le2\sqrt{|D_{2n}|}$. If $n=1+q+q^{2}$ for some prime power $q$, then $\Delta[C_n]=1+q=\frac{1+\sqrt{4n-3}}2$ and $\Delta[D_{2n}]=2+2q=\lceil\!\sqrt{4n}\,\rceil$. Also we calculate the difference weights of all cyclic groups of cardinality $\le 100$, all dihedral groups of cardinality $\le80$, and all Abelian groups of cardinality $<96$.