Discrete Maths question

Question

Show that $S=\{1,3,4,5,9\}$ is a difference set for $\Bbb Z_{11}$.

Identify the design produced from $S$ by the sets of the form $S+i$, $i \in\Bbb Z_{11}$.

Answer 1

Suppose that $S$ is a $(v,k,\lambda)$ difference set in the notation of this article. Then $v=|\Bbb Z_{11}|=11$, and $k=|S|=5$, so the relationship $k^2-k=(v-1)\lambda$ implies that $\lambda=2$: we should be trying to show that each non-zero element of $\Bbb Z_{11}$ can be expressed in the form $s_1-s_2$ with $s_1,s_2\in S$ in exactly two different ways. If all else fails, you can make a table of differences:

$$\begin{array}{c|cc} \text{left}-\text{top}&1&3&4&5&9\\ \hline 1&0&9&8&7&3\\ 3&2&0&10&9&5\\ 4&3&1&0&10&6\\ 5&4&2&1&0&7\\ 9&8&6&5&4&0 \end{array}$$

As you can check by brute force, every non-zero element of $\Bbb Z_{11}$ does indeed appear exactly twice in the table of differences of elements of $S$, so $S$ is an $(11,5,2)$ difference set.

I don’t know in what way you’re supposed to identify the associated block design, but you’ll find it described in the third bullet point under Basic Facts at the link that I gave above.