Question on sets of numbers in "$8_3$ in $PG(2,q)$"...

Question

I was doing brute force numerical experiments and googled some of the results. Surprsingly I found the paper mentioned in the title: "$8_3$ in $PG(2,q)$". In there, out of the blue, sets of numbers are presented $$ \begin{array}{c|rl} - & \{0,1,3\} &\mod 8 \\ PG(2,3) &\{0,1,3,9\} &\mod 13 \\ PG(2,4) & \{0,3,4,9,11\} &\mod 21 \\ PG(2,q)& ...? \end{array} $$ Interestingly all of them were solutions in my brute force numerical search. Can anyone explain how the following sets of numbers are generated?

Answer 1

The authors of [A-EA-DJ] use such sets, called (relative) difference sets as bases for combinatorial schemes of incidences between points and lines. I recall that a subset $D$ of a group $G$ is called a difference set if each non-identity element $g\in G$ can be uniquely written as $g = xy^{−1}$ for some elements $x, y\in D$ [BG]. In 1938 Singer [S] showed that for any prime power $m$ a cyclic group $\Bbb Z_{m_2+m+1}$ contains a difference set of cardinality $m + 1$. He called them perfect difference sets of order $m + 1$) and presented the following table

Thus a set $\{0,1,3\}$ of residues is a difference set for a group $\Bbb Z_7$ but only a relative difference set for a group $\Bbb Z_8$ (in which, for instance, $4$ is not a difference of elements of the set), a set $\{0,1,3,9\}$ of residues is a difference set for a group $\Bbb Z_{13}$, and both sets $\{0,1,4,14,16\}$ and $\{0,3,4,9,11\}$ of residues is a difference set for a group $\Bbb Z_{21}$ (remark that the set $\{0,3,4,9,11\}$ is an image of the set $\{0,1,4,14,16\}$ under a map $x\mapsto 5x+4$ of a ring $\Bbb Z_{21}$).

References

[A-EA-DJ] M.S. Abdul-Elah, M.W. Al-Dhamg, Dieter Jungnickel, $8_3$ in $PG (2, q)$, Arch. math., 49 (1987) 141–150.

[BG] Taras Banakh, Volodymyr Gavrylkiv, Difference bases in cyclic groups.

[S] James Singer, A theorem in finite projective geometry and some applications to number theory, Trans. amer. math. soc. 43:3 (1938), 377–385.