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 [ 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