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Details for Section 7

We provide here more details for Section 7 of the paper. GAP files, containing the generators of the DHOs and their automorphism groups, are linked from within the tables.

Example 7.1

The DHOs of rank 4, mentioned in Example 7.1, can be found in the list of nonisomorphic DHOs.
Direct links: Dim 7: ID 6, ID 10, ID 24. Dim 8: ID 2, ID 4.

Example 7.2

In Example 7.2 we use a simple algorithm which finds for a given bilinear DHO T of rank n the bilinear DHOs S, such that S possesses a hyperplane which induces a subDHO isomorphic to T. We call S a prolongation of T. We apply this to the bilinear DHOs of ranks 4. For any T the number of non-isomorphic S in given dimension are listed in the following table.
The table entries are linked to the corresponding section of the page giving more details of the prolongation DHOs of rank 5.
T\dim S 9 10 11 12 13 14 15
r4_d7_1 0 0 0 1
r4_d7_2 0 0 0 1
r4_d7_3 0 0 0 1
r4_d7_4 0 0 0 1
r4_d7_5 0 0 0 1
r4_d7_6 0 0 0 1
r4_d7_7 0 0 0 0
r4_d8_1 0 0 1 1 1
r4_d8_2 0 0 8 4 1
r4_d8_3 0 0 3 3 1
r4_d8_5 0 3 14 3 1
r4_d8_6 0 2 14 4 1
r4_d8_7 0 0 0 0 0
r4_d8_8 0 0 0 0 0
r4_d8_9 0 1 0 0 0
r4_d8_12 0 12 34 9 1
r4_d8_17 0 0 0 0 0
r4_d9_1 3 2 3 1 1
r4_d9_2 2 19 17 3 1
r4_d9_5 19 4 0 0 0
r4_d9_6 0 0 0 0 0
r4_d9_7 1 2 1 0 0
r4_d10_1 2 3 2 1 1
r4_d10_2 20 24 7 2 1

We also determine the universal cover of prolongation DHOs using the method described in [1, Section 2.5]. These can be found in the table with simply connected DHO of rank 5. We also have put in that table the universal cover of the hyperplane induced subDHO of Example 7.3, the simply connected Extensions of bilinear DHO of rank 4 and the two, not yet appearing, classical simply connected DHO in dimension 15: the Veronesen and the Tanigushi DHO.

The table of non-isomorphic DHO of rank 5 consists of the above simply connected DHO and their quotients.
There are 55 DHO in dimension 10, 647 DHO in dimension 11, 3228 DHO in dimension 12, 154 DHO in dimension 13, 16 DHO in dimension 14 and the well known 4 DHO in dimension 15.

Finaly there is a table identifying the Extensions of bilinear DHO of rank 4 in the table of non-isomorphic DHO of rank 5.

An Example on how to use the tables

Say you want to find the universal covers of prolongations of the bilinear DHO of rank 4 in Dimension 9 with ID 5.
This DHO is identified by r4_d9_5 in the table above. You see in the table that it has 19 prolongations to a DHO of rank 5 in dimension 10 and 4 in dimension 11.
If you click on the 19 you get to the table with details on this prolongation and find in the column U.Cov the universal covers of its 19 prolongations in dimension 10.

Say you are interested to see which universal covers arrise from prolongations but not from extensions.
You look in the table with simply connected DHO of rank 5. All DHO which have no entry in the column "remark" are such universal covers.

Example 7.3

Below the essential data for the DHO of Rank 6 in Example 7.3. The DHO has a hyperplane induced subDHO, isomorphic to the DHO of rank5 in dimension 10 with ID 55.
ID GAP_id |Aut| dim(P) bilinear
1 r6_d11_1 384 10 1

Non-splitting DHO

Among the quotients of DHO of rank 5 there is one non-splitting DHO in dimension 12 (a quotient of the Tanigushi DHO) and 125 non-splitting DHOs in dimension 11 (quotients of the Veronesean and Tanigushi DHO).

References

[1] Ulrich Dempwolff, Doubly transitive dimensional dual hyperovals: universal covers and non-bilinear examples. Adv. Geom. 2019; 19 (3):359–379.

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