REU 2000 at Mount Holyoke College: Difference Sets

This is a summary of results obtained
in the summer 2000 REU on difference sets. Participants: Juliana Belding (Bryn
Mawr College), Elizabeth Carver (Georgia Southern University), Umut Gursel
(Mount Holyoke College), Peter Johnston (Colorado College), Richard Lassow (St.
Olaf College), Jenna Nichols (Mount Holyoke College), Laura Steuble (Mount
Holyoke College). Directed by Harriet Pollatsek, in collaboration with Ken
Smith (Central Michigan University).

Assume D is a (288, 42, 6) difference
set in a group G. If N is a normal subgroup of G then the G/N *intersection numbers* are the sizes of
the intersections of D with the distinct cosets of G modulo N. We write Z_m for
the cyclic group of order m, D_8 for the dihedral group of order 8 and Q_8 for
the quaternion group. We have the following results.

·
The Z_2 intersection numbers are
{24,18}.

·
There are two possibilities for the Z_3
intersection numbers: {18,12,12} or {10,16,16}.

·
There are two possibilities for the Z_4
intersection numbers: {6,12,12,12} or {15,9,9,9}. There are the same two
possibilities for the (Z_2)^2 intersection numbers.

·
There are eight possibilities for the
Z_6 intersection numbers. Those corresponding to the first set of Z_3
intersection numbers are {{12,6}, {6,6}, {6,6}}, {{10,8}, {8,4}, {8,4}},
{{9,9}, {6,6}, {9,3}}, {{7,11}, {7,5}, {8,4}}. Those corresponding to the
second set of Z_3 intersection numbers are {{6,4}, {6,10}, {6,10}}, {{7,3},
{6,10}, {9,7}}, {{8,2}, {8,8},{8,8}}, {{5,5}, {11,5}, {8,8}}.

·
There are four possibilities for Z_9
intersection numbers; they occur only for the first set of Z_3 intersection
numbers. There are the same four possibilities for (Z_3)^2 intersection
numbers. The possibilities are {{10,4,4}, {4,4,4}, {4,4,4}}, {{6,6,6}, {8,2,2},
{4,4,4}}, {{8,8,2}, {4,4,4}, {4,4,4}}, {{6,6,6}, {0,6,6}, {4,4,4}}.

· There
are five possibilities for Z_8 intersection numbers; they occur only for the
even Z_4 intersection numbers. They are {{3, 3}, {9, 3}, {6, 6}, {6, 6}}, {{3,
3}, {8, 4}, {8, 4}, {7, 5}}, {{5, 1}, {8, 4}, {7, 5}, {6, 6}}, {{4, 2}, {8, 4},
{8, 4}, {6, 6}}, {{6, 0}, {6, 6}, {6, 6}, {6, 6}}.

·
There are two possibilities for Z_2 X
Z_4 intersection numbers; there are the same two possibilities for D_8 or
(Z_2)^3 intersection numbers. They are
(grouping by Z_4 intersection numbers) {{6, 0}, {6, 6}, {6, 6}, {6, 6}},
{{9, 3}, {6, 6}, {6, 6}, {6, 6}, {3, 3}}, {{9, 6}, {6, 3}, {6, 3}, {6, 3}}.

·
There are seven possibilities for Q_8
intersection numbers. Those corresponding to the even Z_4 intersection numbers
are {{5, 1}, {8, 4}, {7, 5}, {6, 6}}, {{4, 2}, {8, 4}, {8, 4}, {6, 6}}, {{3,
3}, { 8, 4}, {8, 4}, {7, 5}}. Those corresponding to the odd Z_4 intersection
numbers are {{9, 6}, {6, 3}, {6, 3}, {6, 3}}, {{9, 6}, {7, 2}, {5, 4}, {5, 4}},
{{10, 5}, {6, 3}, {5, 4}, {5, 4}}, {{8, 7}, {7, 2}, {6, 3}, {5, 4}}.

· There
is just one possibility for (Z_2)^4 or (Z_2)^2 X Z_4 or (Z_4)^2 intersection
numbers, namely {6, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 0, 0, 0}.

· There
are many possibilities for Z_{16} intersection numbers, but none for which all
the intersection numbers are even.

· Neither
a (Z_2)^5 nor a Z_{32} image can occur.

In a related project, 272 abelian (96,
20, 4) difference sets were constructed, 32 in Z_4 X (Z_2)^3 X Z_3 and 240 in
(Z_2)^5 X Z_3. All were constructed from Kibler #8 in (Z_2)^4. The construction
strategy is as follows. Regard a (16, 6, 2) difference set D_{16} in G_{16} as
an element of the integral group ring ZG_{16}. Consider

G_{32} = < G_{16}, z >, where we
assume that z^2 is in G_{16}, z is of even order and centralizes D_{16}, and
either z^2=1, or D_{16} = D_{16}^{(-1)}. Let D_{32} = D_{16} + z(G_{16} -
D_{16}) + 2 + 2z \in Z G_{32}. It follows that D_{32} D_{32}^{(-1)} = 16 +
12(G_{16} + zG_{16})= 16 + G_{32}, consistent with D_{32} being a homomorphic
image of a (96, 20, 4) difference set D_{96}. Six of the eight abelian (16, 6,
2) difference sets listed by Kibler (\#3-#8) were used in turn as D_{16}. Using
Maple, all possible pre-images of the corresponding D_{32} in either Z_4 X
(Z_2)^3 X Z_3 or in (Z_2)^5 X Z_3 were tested as possible difference sets
(using the fact that D_{96} is a difference set if and only if \delta =
\chi(D_{96}) satisfies \delta \overline{\delta} = 16 for all nontrivial
characters \chi of G_{96}). For Kibler #3-#8, z^2=1 was used. For Kibler #8,
z^4=1 was also used.

Using this construction strategy, none
of Kibler \#3-#7 produced difference sets in either of these abelian groups of
order 96. It is known that no abelian (96, 20, 4) difference set can exist in a
group whose 2-Sylow subgroup has exponent above 4, so Kibler #1 and #2 were not
considered. It is also known a (96, 20, 4) difference set does exist in (Z_4)^2
X Z_2 X Z_3. Equivalences among these 272 difference sets and between them and
known difference sets (or the corresponding symmetric designs) have not yet
been examined.

[Added summer 2002: Students of Emily Moore of Grinnell College
have used this method to construct four nonabelian (96, 20, 4) difference sets
from Kibler #9. They also found an
error in the construction of the 32 difference sets from Kibler #8 with
z^4=1. In addition, they examined the equivalence question and showed
that all four of the difference sets coming from Kibler #9 are equivalent, as
are the 240 difference sets constructed in 2000 from Kibler #8 with z^2=1. These two classes are inequivalent. They also showed that the McFarland (96, 20,
4) difference set represents a third equivalence class. They are investigating the isomorphism
question for the associated designs.]

The goal in summer 2000 was to
use this strategy to construct a non-abelian (288, 42, 6) difference set from a
(36, 15, 6) difference set. (It is known that an abelian (288, 42, 6)
difference set does not exist.) In this
case we seek a pre-image for D_{72} = D_{36} + z(G_{36} - D_{36}) + 3 + 3z in
G_{72}= < G_{36}, z >. Kibler's list of (36, 15, 6) difference sets was
the point of departure. The first attempts at adapting the Maple programs to
this situation were not successful --- the number of cases to consider was
simply too large to be feasible. But future work will aim to cut the number of
cases down by further analysis and also to write more efficient programs.