A [difference set](http://en.wikipedia.org/wiki/Difference_set) of a group $G$ is a subset $D\subseteq G$ with the property that there exists an integer $\lambda>0$ such that for every non-identity member $g$ of $G$, there exist exactly $\lambda$ ordered pairs $(a,b)\in D\times D$ such that $g=ab^{-1}$. Note that $D=G$ is a difference set with $\lambda=|G|$, and so we typically only consider nontrival difference sets.


[Davis](http://scholar.google.com/scholar?cluster=16666554031641448939) showed that $(\mathbb{Z}/n\mathbb{Z})^2$ admits a nontrivial difference set when $n$ is a power of $2$. Are there any known difference sets when $n$ is odd? Perhaps cyclotomic difference sets?


As far as I know, the [Paley-type construction](http://en.wikipedia.org/wiki/Difference_set#Known_difference_sets) Douglas Zare suggests in the comments (letting $D$ be the set of nonzero perfect squares in $\mathrm{GF}(n^2)$ when $n$ is prime) is only guaranteed to work when $n^2$ is $3\bmod 4$ (which never happens). However, there are hopefully weaker sufficient conditions for $n$ to satisfy, and I think the literature discusses this in the context of "cyclotomic difference sets," but I am not familiar with these results.



Does $(\mathbb{Z}/n\mathbb{Z})^2$ ever admit a difference set when $n$ is odd?
Such difference sets exist. There exist (nontrivial) difference sets with 


$|G| = q^{d+1}[1+(q^{d+1}-1)/(q-1)]$,


$|D| = q^d(q^{d+1}-1)/(q-1)$,


$\lambda = q^d(q^d-1)/(q-1)$,


whenever $q$ is a prime power (R. L. McFarland, A family of difference sets in non-cyclic groups, JCT A, 15 (1973), pp. 1-10). More precisely, such difference sets exist in any abelian group of order $v$ which contains an elementary abelian subgroup of order $q^{d+1}$. 


Take $q=7$ and $d=1$, for instance. This shows that a nontrivial difference set in $(\mathbb{Z}/21\mathbb{Z})^2$ exists.


