International Scholarly Research Notices

Research Article

On the Existence of (𝑣,𝑘,𝜆) Difference Sets with 𝑘<1250 and 𝑘−𝜆 Is a Square

Table 4

Partial results in groups of order 𝑣 by Criterion 2. 𝐶 | 𝐻 | = ⟨ 𝑥 , 𝑦 ∶ 𝑥 𝑞 = 𝑦 2 = [ 𝑥 , 𝑦 ] ⟩ and parameters with asterisk indicate new results. No difference set image in 𝐻 implies no diff. set image in Dihedral group of same order. ? means the number of groups of order 𝑣 is unknown.
( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 | 𝐻 | Factoring of 𝑝 in ℤ [ 𝜁 | 𝐻 / ⟨ 𝑔 ⟩ | ] No. of groups of order 𝑣 No. of groups ruled outSolutions in 𝐻
1(78, 22, 6) 4 2 26 2 6 ≡ − 1 ( m o d 1 3 ) 64 − 4 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2(204, 29, 4) 5 5 34 5 8 ≡ − 1 ( m o d 1 7 ) 1210 − 5 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 ∗ (1140, 68, 4) 8 2 38 2 9 ≡ − 1 ( m o d 1 9 ) 4129 − 8 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 ∗ (396, 80, 16) 8 2 66 2 5 ≡ − 1 ( m o d 𝑏 )
𝑏 = 1 1 , 3 3 		3015None in 𝐻 / ⟨ 𝑔 ⟩
5 ∗ (300, 92, 48) 8 2 50 2 1 0 ≡ − 1 ( m o d 2 5 ) 499 − 8 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
6 ∗ (980, 89, 8) 9 3 14 3 3 ≡ − 1 ( m o d 7 ) 3431 − 9 + 7 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
7 ∗ (456, 105, 24) 9 3 38 3 9 ≡ − 1 ( m o d 1 9 ) 5439 − 9 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
8 ∗ (1856, 106, 6) 10 2, 5 58 𝑎 ≡ − 1 ( m o d 2 9 )
𝑎 = 2 1 4 , 5 7 		16301387 − 1 0 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
9 ∗ (7504, 123, 2) 11 11 134 1 1 3 3 ≡ − 1 ( m o d 6 7 ) ?? − 1 1 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 0 ∗ (3876, 125, 4) 11 11 34 1 1 8 ≡ − 1 ( m o d 1 7 ) 4034 − 1 1 + 8 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 1 ∗ (2870, 152, 8) 12 2, 3 82 𝑎 ≡ − 1 ( m o d 4 1 )
𝑎 = 2 1 0 , 3 4 		128 − 1 2 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 2 ∗ (666, 210, 66) 12 2, 3 74 𝑎 ≡ − 1 ( m o d 3 7 )
𝑎 = 2 1 8 , 3 9 		1810 − 1 2 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 3 ∗ (610, 232, 88) 12 2, 3 122 𝑎 ≡ − 1 ( m o d 6 1 )
𝑎 = 2 3 0 , 3 5 		64 − 1 2 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 4 ∗ (14536, 171, 2) 13 13 46Remark 2.3?? − 1 3 + 8 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 5 ∗ (7440, 173, 4) 13 13 62 1 3 1 5 ≡ − 1 ( m o d 3 1 ) ?? − 1 3 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 6 ∗ (5076, 175, 6) 13 13 94 1 3 2 3 ≡ − 1 ( m o d 4 7 ) ?? − 1 3 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 7 ∗ (2716, 181, 12) 13 13 194 1 3 4 8 ≡ − 1 ( m o d 9 7 ) 11 9 − 1 3 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 8 ∗ (19504, 198, 2) 14 2, 7 106 𝑎 ≡ − 1 ( m o d 5 3 )
𝑎 = 2 2 6 , 7 1 3 		?? − 1 4 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 9 ∗ (1696, 226, 30) 14 2, 7 212 2 2 6 ≡ − 1 ( m o d 5 3 )
7 1 3 ≡ − 1 ( m o d 𝑏 )
𝑏 = 5 3 , 1 0 6 , 2 1 2 		235194 1 4 + ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
2 0 ∗ (8856, 231, 6) 15 3, 5 82 𝑎 ≡ − 1 ( m o d 4 1 )
𝑎 = 3 4 , 5 2 0 		?? − 1 5 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 1 ∗ (1508, 275, 50) 15 3, 5 58 𝑎 ≡ − 1 ( m o d 2 9 )
𝑎 = 3 1 4 , 5 7 		15 11 − 1 5 + 1 0 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 2 ∗ (976, 351, 126) 15 3, 5 122 𝑎 ≡ − 1 ( m o d 6 1 )
𝑎 = 3 5 , 5 1 5 		51 14 − 1 5 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 3 ∗ (3256, 280, 24) 16 2 74 2 1 8 ≡ − 1 ( m o d 3 7 ) ?? − 1 6 + 8 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 4 ∗ (1036, 460, 204) 16 2 74 2 1 8 ≡ − 1 ( m o d 3 7 ) 119 1 6 + 6 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ ; 6 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ + 8 ( 1 + 𝑥 + 𝑦 − 𝑥 𝑦 )
2 5 ∗ (21390, 293, 4) 17 17 62 1 7 1 5 ≡ − 1 ( m o d 3 1 ) 3616 − 1 7 + 1 0 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 6 ∗ (52976, 326, 2) 18 2, 3 86 3 2 1 ≡ − 1 ( m o d 4 3 ) See Remark 2.2 for ⟨ 2 ⟩ ?? − 1 8 + 8 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 7 ∗ (18096, 330, 6) 18 2, 3 58 𝑎 1 4 ≡ − 1 ( m o d 2 9 )
𝑎 = 2 , 3 		?? − 1 8 + 1 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 8 ∗ (1776, 426, 102) 18 2, 3 74 𝑎 ≡ − 1 ( m o d 3 7 )
𝑎 = 2 1 8 , 3 9 		260170 − 1 8 + 1 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 9 ∗ (65704, 363, 2) 19 19 382 1 9 9 5 ≡ − 1 ( m o d 1 9 1 ) ? ? − 1 9 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 0 ∗ (22388, 367, 6) 19 19 386 1 9 9 6 ≡ − 1 ( m o d 1 9 3 ) 15 11 − 1 9 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 1 ∗ (13728, 371, 10) 19 19 26 1 9 6 ≡ − 1 ( m o d 1 3 ) ? ? − 1 9 + 1 5 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
3 2 ∗ (7960, 379, 18) 19 19 398 1 9 9 ≡ − 1 ( m o d 1 9 9 ) ?? − 1 9 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 3 ∗ (5084, 391, 30) 19 19 82 1 9 2 0 ≡ − 1 ( m o d 4 1 ) 11 9 − 1 9 + 1 0 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 4 ∗ (2256, 451, 90) 19 19 94 1 9 2 3 ≡ − 1 ( m o d 4 7 ) ? ? − 1 9 + 1 0 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 5 ∗ (40704, 404, 4) 20 2, 5 106 𝑎 2 6 ≡ − 1 ( m o d 5 3 )
𝑎 = 2 , 5 		?? − 2 0 + 8 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 6 ∗ (16770, 410, 10) 20 2, 5 86 𝑎 ≡ − 1 ( m o d 4 3 )
𝑎 = 2 7 , 5 2 1 		48 24 − 2 0 + 1 0 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 7 ∗ (1830, 590, 190) 20 2, 5 122 𝑎 ≡ − 1 ( m o d 6 1 )
𝑎 = 2 3 0 , 5 1 5 		18 8 − 2 0 + 1 0 ⟨ 𝑥 ⟩ i n 𝐻 / ⟨ 𝑔 ⟩
3 8 ∗ (49369, 445, 4) 21 3, 7 466 𝑎 ≡ − 1 ( m o d 2 3 3 )
𝑎 = 3 1 1 6 , 7 5 8 		1511 − 2 1 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 9 ∗ (17064, 453, 12) 21 3, 7 158 𝑎 3 9 ≡ − 1 ( m o d 7 9 )
𝑎 = 3 , 7 		?? − 2 1 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 0 ∗ (14756, 455, 14) 21 3, 7 34 𝑎 8 ≡ − 1 ( m o d 1 7 )
𝑎 = 3 , 7 		2723 − 2 1 + 1 4 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
4 1 ∗ (10604, 461, 20) 21 3, 7 482 𝑎 ≡ − 1 ( m o d 2 4 1 )
𝑎 = 3 6 0 , 7 1 2 0 		119 − 2 1 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 2 ∗ (7380, 471, 30) 21 3, 7 82 𝑎 ≡ − 1 ( m o d 4 1 )
𝑎 = 3 4 , 7 2 0 		14989 − 2 1 + 1 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 3 ∗ (5336, 485, 44) 21 3, 7 46Remark 2.3? ? − 2 1 + 1 1 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
4 4 ∗ (3128, 531, 90) 21 3, 7 46Remark 2.3? ? − 2 1 + 1 2 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
4 5 ∗ (2408, 581, 140) 21 3, 7 86 𝑎 ≡ − 1 ( m o d 4 3 )
𝑎 = 3 2 1 , 7 3 		? ? − 2 1 + 1 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 6 ∗ (70890, 533, 4) 23 23 278 2 3 2 3 ≡ − 1 ( m o d 1 3 9 ) 24 16 − 2 3 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 7 ∗ (47616, 535, 8) 23 23 62 2 3 5 ≡ − 1 ( m o d 3 1 ) ? ? − 2 3 + 1 8 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 8 ∗ (13776, 551, 22) 23 23 82 2 3 5 ≡ − 1 ( m o d 4 1 ) ? ? − 2 3 + 7 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
4 9 ∗ (166754, 578, 2) 24 2, 3 86 𝑎 ≡ − 1 ( m o d 4 3 )
𝑎 = 2 7 , 3 2 1 		12 8 − 2 4 + 1 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
5 0 ∗ (56358, 582, 6) 24 2, 3 202 𝑎 5 0 ≡ − 1 ( m o d 1 0 1 )
𝑎 = 2 , 3 		? ? − 2 4 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
5 1 ∗ (10388, 612, 36) 24 2, 3 106 𝑎 2 6 ≡ − 1 ( m o d 5 3 )
𝑎 = 2 , 3 		34 28 − 2 4 + 1 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
5 2 ∗ (4844, 668, 92) 24 2, 3 346 𝑎 8 6 ≡ − 1 ( m o d 1 7 3 )
𝑎 = 2 , 3 		11 9 − 2 4 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
5 3 ∗ (3690, 714, 138) 24 2, 3 82 𝑎 ≡ − 1 ( m o d 4 1 )
𝑎 = 2 1 0 , 3 4 		30 20 − 2 4 + 1 8 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
5 4 ∗ (3172, 756, 180) 24 2, 3 122 𝑎 ≡ − 1 ( m o d 6 1 )
𝑎 = 2 3 0 , 3 5 		15 11 2 4 + 6 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ , 6 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ + 1 2 ( 1 + 𝑥 + 𝑦 − 𝑥 𝑦 )
5 5 ∗ (2332, 1036, 460) 24 2, 3 106 𝑎 2 6 ≡ − 1 ( m o d 5 3 )
𝑎 = 2 , 3 		11 9 − 2 4 + 2 0 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
5 6 ∗ (196252, 627, 2) 25 5326 5 2 7 ≡ − 1 ( m o d 1 6 3 ) ? ? − 2 5 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
5 7 ∗ (66256, 631, 6) 25 582 5 1 0 ≡ − 1 ( m o d 4 1 ) ? ? − 2 5 + 1 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
5 8 ∗ (50008, 633, 8) 25 594 5 2 3 ≡ − 1 ( m o d 4 7 ) ? ? − 2 5 + 1 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
5 9 ∗ (17524, 649, 24) 25 5674 5 5 6 ≡ − 1 ( m o d 3 3 7 ) 15 11 − 2 5 + 2 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
6 0 ∗ (14280, 655, 30) 25 534 5 8 ≡ − 1 ( m o d 1 7 ) ? ? − 2 5 + 2 0 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
6 1 ∗ (11040, 665, 40) 25 512 5 ≡ − 1 ( m o d 𝑏 ) 𝑏 = 2 , 3 ? ? − 2 5 + 1 1 5 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
6 2 ∗ (11040, 665, 40) 25 546Remark 2.3 ? ? − 2 5 + 1 5 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
6 3 ∗ (5104, 729, 104) 25 558 5 7 ≡ − 1 ( m o d 2 9 ) ? ? − 2 5 + 1 3 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
6 4 ∗ (2812, 937, 312) 25 574 5 1 8 ≡ − 1 ( m o d 3 7 ) 11 9 − 2 5 + 1 3 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩