# Number of cyclic difference sets

A subset $D=\{a_1,\ldots,a_k\}$ of $\mathbb{Z}/v\mathbb{Z}$ is said to be a $(v,k,\lambda)$-cycic difference set if for each nonzero $b\in\mathbb{Z}/v\mathbb{Z}$, there are exactly $\lambda$ ordered pairs $(a_s,a_t)\in D^2$ such that $a_s-a_t=b$. For a $(v,k,\lambda)$-difference set $D$, $k-\lambda$ is called the order of $D$.

Let $N_v$ be the number of different (even they are equivalent, different subsets of $\mathbb{Z}/v\mathbb{Z}$ viewd as different) cyclic difference sets in $\mathbb{Z}/v\mathbb{Z}$, and $N_{v,n}$ be the number of different $(v,k,\lambda)$-cyclic difference sets of order $n$. Are there any nontrivial upper bounds known for $N_v$ or $N_{v,n}$?

• What is n? If it is $k-\lambda$, then your $N_v$ and $N_{v,n}$ appear to be the same. I am not aware of results for cyclic groups, but Muzychuk has constructed exponentially many equivalence classes of difference sets in certain elementary abelian groups. – Padraig Ó Catháin Oct 4 '14 at 4:41
• @PadraigÓCatháin Yes, $n$ means the order $k-\lambda$. Do you mean given any $v$, $N_v$ should be equal to $N_{v,n}$ for some $n$? And what is the title of Muzychuk's paper you mentioned?:) – Binzhou Xia Oct 4 '14 at 7:04
• I changed the wording of your question slightly to make it more clear. The title of Muzychuk's paper is 'On skew Hadamard difference sets'. – Padraig Ó Catháin Oct 4 '14 at 7:24