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 3

Parameter sets that do not exist by Criterion 2. 𝐶 | 𝐻 | = ⟨ 𝑥 , 𝑦 ∶ 𝑥 𝑞 = 𝑦 2 = [ 𝑥 , 𝑦 ] ⟩ and parameters with asterisk indicate new results. No diff. set image in 𝐻 implies no difference set image in Dihedral group of same order.
( 𝑣 , 𝑘 , 𝜆 ) 𝑚 𝑝 | 𝐻 | Factoring of 𝑝 in ℤ [ 𝜁 | 𝐻 / ⟨ 𝑔 ⟩ | ] No. of groups of order 𝑣 Solutions in 𝐻
1(56, 11, 2)3 314 3 3 ≡ − 1 ( m o d 7 ) 13 − 3 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2(154, 18, 2)4 222 2 5 ≡ − 1 ( m o d 1 1 ) 4 − 4 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3(66, 26, 10) 4 2 22 2 5 ≡ − 1 ( m o d 1 1 ) 2 4 + ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ ; ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ + 2 ( 1 + 𝑥 + 𝑦 − 𝑥 𝑦 )
4(112, 37, 2) 5 5 14 5 3 ≡ − 1 ( m o d 7 ) 43 − 5 + 3 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
5 ∗ (690, 53, 4) 7 7 10 7 2 ≡ − 1 ( m o d 5 ) 8 − 7 + 6 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
6(496, 55, 6) 7 7 627 factors trivially see [21] 42 − 7 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
7 ∗ (306, 61, 12) 7 7 34 7 8 ≡ − 1 ( m o d 1 7 ) 10 − 7 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
8 ∗ (2146, 66, 2) 8 2 74 2 1 8 ≡ − 1 ( m o d 3 7 ) 4 − 8 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
9 ∗ (806, 70, 6) 8 2 26 2 6 ≡ − 1 ( m o d 1 3 ) 4 − 8 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 0 ∗ (430, 78, 14) 8 2 86 2 7 ≡ − 1 ( m o d 4 3 ) 4 − 8 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 1 ∗ (370, 82, 18) 8 2 74 2 1 8 ≡ − 1 ( m o d 3 7 ) 4 8 + ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ ; ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ + 4 ( 1 + 𝑥 + 𝑦 − 𝑥 𝑦 )
1 2 ∗ (266, 106, 42) 8 2 38 2 9 ≡ − 1 ( m o d 1 9 ) 4 − 8 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 3 ∗ (3404, 83, 2) 9 3 46Remark 2.3 11 − 9 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 4 ∗ (714, 93, 12) 9 3 34 3 8 ≡ − 1 ( m o d 1 7 ) 12 − 9 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 5 ∗ (2668, 127, 6) 11 11 46Remark 2.3 11 − 1 1 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 6 ∗ (1704, 131, 10) 11 11 142 1 1 3 5 ≡ − 1 ( m o d 7 1 ) 39 − 1 1 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 7 ∗ (1450, 162, 18) 12 2,3 58 𝑎 1 4 ≡ − 1 ( m o d 2 9 )
𝑎 = 2 , 3 		10 − 1 2 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
1 8 ∗ (760, 253, 84) 13 13 38 1 3 9 ≡ − 1 ( m o d 1 9 ) 39 − 1 3 + 7 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
1 9 ∗ (13054, 229, 4) 15 3,5 122 𝑎 5 ≡ − 1 ( m o d 6 1 )
𝑎 = 3 , 5 		4 − 1 5 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 0 ∗ (4064, 239, 14) 15 3, 5 254 𝑎 ≡ − 1 ( m o d 1 2 7 )
𝑎 = 3 6 3 , 5 2 1 		195 − 1 5 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 1 ∗ (3268, 243, 18) 15 3,5 86 𝑎 2 1 ≡ − 1 ( m o d 4 3 )
𝑎 = 3 , 5 		9 − 1 5 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 2 ∗ (2278, 253, 28) 15 3,5 134 𝑎 1 1 ≡ − 1 ( m o d 6 7 )
𝑎 = 3 , 5 		4 − 1 5 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 3 ∗ (1886, 261, 36) 15 3,5 46Remark 2.3 4 − 1 5 + 1 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 4 ∗ (1406, 281, 56) 15 3,5 74 𝑎 ≡ − 1 ( m o d 3 7 )
𝑎 = 3 9 , 5 1 8 		4 − 1 5 + 8 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 5 ∗ (1054, 325, 100) 15 3,5 34 𝑎 8 ≡ − 1 ( m o d 1 7 )
𝑎 = 3 , 5 		4 − 1 5 + 1 0 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
2 6 ∗ (918, 393, 168) 15 3,5 34 𝑎 8 ≡ − 1 ( m o d 1 7 )
𝑎 = 3 , 5 		30 − 1 5 + 1 2 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
2 7 ∗ (902, 425, 200) 15 3,5 82 𝑎 ≡ − 1 ( m o d 4 1 )
𝑎 = 3 4 , 5 1 0 		4 1 5 + 5 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩
2 8 ∗ (33154, 258, 2) 16 2 274 2 3 4 ≡ − 1 ( m o d 1 3 7 ) 10 − 1 6 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
2 9 ∗ (11398, 262, 6) 16 2 278 2 6 9 ≡ − 1 ( m o d 1 3 9 ) 4 − 1 6 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 0 ∗ (2466, 290, 34) 16 2 274 2 3 4 ≡ − 1 ( m o d 1 3 7 ) 10 1 6 + ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ ; ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ + 8 ( 1 + 𝑥 + 𝑦 − 𝑥 𝑦 )
3 1 ∗ (1660, 316, 60) 16 2 166 2 4 1 ≡ − 1 ( m o d 8 3 ) 11 − 1 6 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 2 ∗ (1066, 426, 170) 16 2 82 2 1 0 ≡ − 1 ( m o d 4 1 ) 4 1 6 + 5 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ ; 5 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ + 8 ( 1 + 𝑥 + 𝑦 − 𝑥 𝑦 )
3 3 ∗ (7526, 301, 12) 17 17 106 1 7 1 3 ≡ − 1 ( m o d 5 3 ) 4 − 1 7 + 6 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 4 ∗ (5796, 305, 16) 17 17 46 1 7 1 1 ≡ − 1 ( m o d 2 3 ) 111 − 1 7 + 1 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 5 ∗ (20758, 408, 8) 20 2,5 214 𝑎 5 3 ≡ − 1 ( m o d 1 0 7 )
𝑎 = 2 , 5 		4 − 2 0 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 6 ∗ (7474, 424, 24) 20 2,5 74 𝑎 1 8 ≡ − 1 ( m o d 3 7 )
𝑎 = 2 , 5 		4 − 2 0 + 1 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 7 ∗ (5038, 438, 38) 20 2,5 458 𝑎 ≡ − 1 ( m o d 2 2 9 )
𝑎 = 2 3 8 , 5 5 7 		4 − 2 0 + 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
3 8 ∗ (2014, 550, 150) 20 2,5 106 𝑎 2 6 ≡ − 1 ( m o d 5 3 )
𝑎 = 2 , 5 		4 2 0 + 5 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ ; 5 ⟨ 𝑥 ⟩ ⟨ 𝑦 ⟩ + 1 0 ( 1 + 𝑥 + 𝑦 − 𝑥 𝑦 )
3 9 ∗ (1918, 568, 168) 20 2,5 274 𝑎 ≡ − 1 ( m o d 1 3 7 )
𝑎 = 2 3 4 , 5 6 8 		4 2 0 + 4 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 0 ∗ (24346, 541, 12) 23 23 94 2 3 2 3 ≡ − 1 ( m o d 4 7 ) 8 − 2 3 + 1 2 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 1 ∗ (34282, 586, 10) 24 2, 3 122 𝑎 ≡ − 1 ( m o d 6 1 )
𝑎 = 2 3 0 , 3 5 		4 − 2 4 + 1 0 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩
4 2 ∗ (20770, 645, 20) 25 5134 5 1 1 ≡ − 1 ( m o d 6 7 ) 12 − 2 5 + 1 0 ⟨ 𝑥 ⟩ in 𝐻 / ⟨ 𝑔 ⟩