Constructing $\lambda$-difference sets

Question

Given a set say $A=\{0,1,4,16,r\}$ which is a subset of $\mathbb{Z}_{21}$. How do I find $r$, such that $A$ is a $\lambda$-difference set for some $\lambda$?

Is there some methodical way to solve problems like this?

Answer 1

Since your set has 5 elements, you can create $5\cdot 4=20$ differences. Therefore your $\lambda$ is equal to 1. Your set already covers differences 1,3,4,5,6,9,12,15,16,17,18 and 20 in $\mathbb Z_{21}$. So $r$ should be at "distance" two from one of the elements of $A$. The possibilities are 2,3,6,14,18,19,20. We can exclude 3,6,18 and 20, since they difference with zero will give some differences more than once. So you have to consider numbers 2,14,19. Since $2-0=4-2$ you can't put $r=2$. On the other hand $4-1=1-19$, so 19 is also excluded. Therefore the last option is $r=14$. If you check all the differences, then you figure out that it is a good choice.