The DHO of rank 2 is the dual of the hyperoval thus unique.
The DHOs of rank 3 were completely classified by Del Fra [2].
The table of DHOs of rank 4 bases on [1]. There the DHOs of rank 4 and dimension smaller than 9 are completely classified, as well as the bilinear DHO of rank 4 for any dimension.
The table of DHOs rank 5 mostly bases on examples given in
the paper. It aims no classification and thus not even lists all known DHO of rank 5.
The references above are found in the bibliography of the paper.
ID | GAP_id | |Aut| | dim(P) | splitting | bilinear | univ.cov. | covers in dim 4 |
---|---|---|---|---|---|---|---|
1 | r2_d3_1 | 24 | 1 | 1 | 1 | s.c. | s.c. |
ID | GAP_id | |Aut| | dim(P) | splitting | bilinear | univ.cov. | covers in dim 6 |
---|---|---|---|---|---|---|---|
1 | r3_d5_1 | 168 | 5 | 1 | 1 | s.c. | s.c. |
ID | GAP_id | |Aut| | dim(P) | splitting | bilinear | univ.cov. | covers in dim 7 |
---|---|---|---|---|---|---|---|
1 | r3_d6_1 | 1344 | 3 | 1 | 1 | s.c. | s.c. |
2 | r3_d6_2 | 168 | 6 | 1 | 0 | s.c. | s.c. |
ID | GAP_id | |Aut| | dim(P) | splitting | bilinear | univ.cov. | covers in dim 8 |
---|---|---|---|---|---|---|---|
1 | r4_d7_1 | 2688 | 4 | 1 | 1 | r4_d10_2 | r4_d8_3 |
2 | r4_d7_2 | 960 | 7 | 1 | 1 | s.c. | s.c. |
3 | r4_d7_3 | 384 | 5 | 1 | 1 | r4_d8_2 | r4_d8_2 |
4 | r4_d7_4 | 288 | 7 | 1 | 1 | r4_d9_6 | r4_d8_6 |
5 | r4_d7_5 | 112 | 7 | 1 | 1 | s.c. | s.c. |
6 | r4_d7_6 | 96 | 6 | 1 | 1 | s.c. | s.c. |
7 | r4_d7_7 | 64 | 7 | 1 | 1 | r4_d8_8 | r4_d8_8 |
8 | r4_d7_8 | 48 | 7 | 1 | 0 | s.c. | s.c. |
9 | r4_d7_9 | 48 | 7 | 1 | 0 | s.c. | s.c. |
10 | r4_d7_10 | 48 | 6 | 1 | 0 | s.c. | s.c. |
11 | r4_d7_11 | 42 | 7 | 1 | 0 | s.c. | s.c. |
12 | r4_d7_12 | 36 | 7 | 1 | 0 | s.c. | s.c. |
13 | r4_d7_13 | 24 | 7 | 1 | 0 | s.c. | s.c. |
14 | r4_d7_14 | 24 | 7 | 1 | 0 | s.c. | s.c. |
15 | r4_d7_15 | 24 | 7 | 1 | 0 | r4_d8_13 | r4_d8_13 |
16 | r4_d7_16 | 24 | 7 | 1 | 0 | r4_d8_14 | r4_d8_14 |
17 | r4_d7_17 | 20 | 7 | 1 | 0 | s.c. | s.c. |
18 | r4_d7_18 | 18 | 7 | 1 | 0 | s.c. | s.c. |
19 | r4_d7_19 | 16 | 7 | 1 | 0 | s.c. | s.c. |
20 | r4_d7_20 | 12 | 7 | 1 | 0 | s.c. | s.c. |
21 | r4_d7_21 | 12 | 7 | 1 | 0 | s.c. | s.c. |
22 | r4_d7_22 | 9 | 7 | 1 | 0 | s.c. | s.c. |
23 | r4_d7_23 | 8 | 7 | 1 | 0 | s.c. | s.c. |
24 | r4_d7_24 | 8 | 6 | 1 | 0 | s.c. | s.c. |
25 | r4_d7_25 | 6 | 7 | 1 | 0 | s.c. | s.c. |
26 | r4_d7_26 | 6 | 7 | 1 | 0 | s.c. | s.c. |
27 | r4_d7_27 | 6 | 7 | 1 | 0 | s.c. | s.c. |
28 | r4_d7_28 | 6 | 7 | 1 | 0 | s.c. | s.c. |
29 | r4_d7_29 | 6 | 7 | 1 | 0 | s.c. | s.c. |
30 | r4_d7_30 | 6 | 7 | 1 | 0 | s.c. | s.c. |
31 | r4_d7_31 | 4 | 7 | 1 | 0 | s.c. | s.c. |
32 | r4_d7_32 | 4 | 7 | 1 | 0 | s.c. | s.c. |
33 | r4_d7_33 | 3 | 7 | 1 | 0 | s.c. | s.c. |
34 | r4_d7_34 | 3 | 7 | 1 | 0 | s.c. | s.c. |
35 | r4_d7_35 | 2 | 7 | 1 | 0 | s.c. | s.c. |
36 | r4_d7_36 | 2 | 7 | 1 | 0 | s.c. | s.c. |
37 | r4_d7_37 | 2 | 7 | 1 | 0 | s.c. | s.c. |
ID | GAP_id | |Aut| | dim(P) | splitting | bilinear | univ.cov. | covers in dim 9 |
---|---|---|---|---|---|---|---|
1 | r4_d8_1 | 5760 | 4 | 1 | 1 | r4_d10_1 | r4_d9_1 |
2 | r4_d8_2 | 384 | 6 | 1 | 1 | s.c. | s.c. |
3 | r4_d8_3 | 384 | 5 | 1 | 1 | r4_d10_2 | r4_d9_2 |
4 | r4_d8_4 | 288 | 6 | 1 | 0 | s.c. | s.c. |
5 | r4_d8_5 | 96 | 8 | 1 | 1 | r4_d9_7 | r4_d9_7 |
6 | r4_d8_6 | 96 | 8 | 1 | 1 | r4_d9_6 | r4_d9_6 |
7 | r4_d8_7 | 96 | 8 | 1 | 1 | r4_d9_6 | r4_d9_6 |
8 | r4_d8_8 | 64 | 8 | 1 | 1 | s.c. | s.c. |
9 | r4_d8_9 | 64 | 8 | 1 | 1 | r4_d9_7 | r4_d9_7 |
10 | r4_d8_10 | 60 | 8 | 1 | 0 | r4_d10_3 | r4_d9_3 |
11 | r4_d8_11 | 36 | 8 | 1 | 0 | s.c. | s.c. |
12 | r4_d8_12 | 32 | 8 | 1 | 1 | s.c. | s.c. |
13 | r4_d8_13 | 24 | 8 | 1 | 0 | s.c. | s.c. |
14 | r4_d8_14 | 24 | 8 | 1 | 0 | s.c. | s.c. |
15 | r4_d8_15 | 18 | 8 | 1 | 0 | s.c. | s.c. |
16 | r4_d8_16 | 16 | 8 | 1 | 0 | s.c. | s.c. |
17 | r4_d8_17 | 16 | 8 | 1 | 1 | s.c. | s.c. |
18 | r4_d8_18 | 12 | 8 | 1 | 0 | s.c. | s.c. |
19 | r4_d8_19 | 12 | 8 | 1 | 0 | s.c. | s.c. |
20 | r4_d8_20 | 8 | 8 | 1 | 0 | s.c. | s.c. |
21 | r4_d8_21 | 8 | 8 | 1 | 0 | s.c. | s.c. |
22 | r4_d8_22 | 8 | 8 | 1 | 0 | s.c. | s.c. |
23 | r4_d8_23 | 6 | 8 | 1 | 0 | s.c. | s.c. |
24 | r4_d8_24 | 4 | 8 | 1 | 0 | s.c. | s.c. |
25 | r4_d8_25 | 1 | 8 | 1 | 0 | s.c. | s.c. |
26 | r4_d8_26 | 1 | 8 | 1 | 0 | s.c. | s.c. |
ID | GAP_id | |Aut| | dim(P) | splitting | bilinear | univ.cov. | covers in dim 10 |
---|---|---|---|---|---|---|---|
1 | r4_d9_1 | 11520 | 5 | 1 | 1 | r4_d10_1 | r4_d10_1 |
3 | r4_d9_3 | 120 | 9 | 1 | 0 | r4_d10_3 | r4_d10_3 |
6 | r4_d9_6 | 288 | 9 | 1 | 1 | s.c. | s.c. |
7 | r4_d9_7 | 192 | 9 | 1 | 1 | s.c. | s.c. |
2 | r4_d9_2 | 768 | 6 | 1 | 1 | r4_d10_2 | r4_d10_2 |
5 | r4_d9_5 | 2688 | 8 | 1 | 1 | s.c. | s.c. |
4 | r4_d9_4 | 12 | 9 | 1 | 0 | r4_d10_4 | r4_d10_4 |
ID | GAP_id | |Aut| | dim(P) | splitting | bilinear | univ.cov. | covers in dim 11 |
---|---|---|---|---|---|---|---|
1 | r4_d10_1 | 322560 | 6 | 1 | 1 | s.c. | s.c. |
2 | r4_d10_2 | 21504 | 7 | 1 | 1 | s.c. | s.c. |
3 | r4_d10_3 | 20160 | 10 | 1 | 0 | s.c. | s.c. |
4 | r4_d10_4 | 1344 | 10 | 1 | 0 | s.c. | s.c. |