Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $S+i$, $i \in\Bbb Z_{11}$.

share|improve this question
    
Please don’t remove the question. –  Brian M. Scott Jan 7 '13 at 17:28
    
I wasn't trying to remove it, I was going to change the question so that the right mathematical symbols were on the question. Thanks for beating me to it. –  user55474 Jan 7 '13 at 17:30

1 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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.