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. | 
| ID | GAP_id | |Aut| | dim(P) | splitting | bilinear | univ.cov. | covers in dim 15 | 
|---|---|---|---|---|---|---|---|
| 1 | r5_d14_1 | 368640 | 9 | 1 | 1 | r5_d15_1 | r5_d15_1 | 
| 2 | r5_d14_2 | 23040 | 10 | 1 | 0 | r5_d15_1 | r5_d15_1 | 
| 3 | r5_d14_3 | 24576 | 10 | 1 | 1 | r5_d15_2 | r5_d15_2 | 
| 4 | r5_d14_4 | 23040 | 10 | 1 | 1 | r5_d15_2 | r5_d15_2 | 
| 5 | r5_d14_5 | 768 | 11 | 1 | 0 | r5_d15_2 | r5_d15_2 | 
| 6 | r5_d14_6 | 23040 | 10 | 1 | 0 | r5_d15_2 | r5_d15_2 | 
| 7 | r5_d14_7 | 1920 | 14 | 1 | 0 | r5_d15_3 | r5_d15_3 | 
| 8 | r5_d14_8 | 720 | 14 | 1 | 0 | r5_d15_3 | r5_d15_3 | 
| 9 | r5_d14_9 | 192 | 14 | 1 | 0 | r5_d15_4 | r5_d15_4 | 
| 10 | r5_d14_10 | 720 | 14 | 1 | 0 | r5_d15_4 | r5_d15_4 | 
| 11 | r5_d14_11 | 48 | 14 | 1 | 0 | r5_d15_4 | r5_d15_4 | 
| 12 | r5_d14_12 | 48 | 14 | 1 | 0 | r5_d15_4 | r5_d15_4 | 
| 13 | r5_d14_13 | 720 | 14 | 1 | 0 | r5_d15_4 | r5_d15_4 | 
| 14 | r5_d14_14 | 6144 | 13 | 1 | 0 | s.c. | s.c. | 
| 15 | r5_d14_15 | 9216 | 13 | 1 | 0 | s.c. | s.c. | 
| 16 | r5_d14_16 | 36864 | 12 | 1 | 0 | s.c. | s.c. | 
| ID | GAP_id | |Aut| | dim(P) | splitting | bilinear | univ.cov. | covers in dim 16 | 
|---|---|---|---|---|---|---|---|
| 1 | r5_d15_1 | 319979520 | 10 | 1 | 1 | s.c. | s.c. | 
| 2 | r5_d15_2 | 10321920 | 11 | 1 | 1 | s.c. | s.c. | 
| 3 | r5_d15_3 | 9999360 | 15 | 1 | 0 | s.c. | s.c. | 
| 4 | r5_d15_4 | 322560 | 15 | 1 | 0 | s.c. | s.c. |