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

Table 5

Partial results in groups of order 𝑣 by Criterion 3. Parameters with asterisk indicate new results.? means the number of groups of order 𝑣 is unknown.

	(𝑣, 𝑘, 𝜆)	𝑚	𝑝	Factoring of 𝑝in ℤ[𝜁𝑞]	No. of groups of order 𝑣	No. of groups ruled out	Solutions in 𝐻
1	(40, 13, 4)	3	3	3^2≡−1(mod𝑏); 𝑏=5,10	14	10	3+𝐻, |𝐻|=10
2^∗	(400, 57, 8)	7	7	7^2≡−1(mod𝑏); 𝑏=5,10	221	166	7+5𝐻, |𝐻|=10
3^∗	(280, 63, 14)	7	7	7^2≡−1(mod𝑏); 𝑏=5,10	40	30	−7+7𝐻, |𝐻|=10
4^∗	(220, 73, 24)	7	7	7^2≡−1(mod𝑏); 𝑏=5,10	15	7	−7+8𝐻, |𝐻|=10
5^∗	(820, 91, 10)	9	3	3^2≡−1(mod𝑏); 𝑏=5,10	20	7	−9+10𝐻, |𝐻|=10
6^∗	(540, 99, 18)	9	3	3^2≡−1(mod𝑏); 𝑏=5,10	119	56	9+9𝐻, |𝐻|=10
7^∗	(3876, 125, 4)	11	11	11≡−1(mod𝑏); 𝑏=3,6,12	40	32	11+19𝐻, |𝐻|=6
8^∗	(1464, 133, 12)	11	11	11^2≡−1(mod61)	61	30	11+𝐻, |𝐻|=122
9^∗	(988, 141, 30)	11	11	11^6≡−1(mod𝑏); 𝑏=13,26	11	5	11+5𝐻, |𝐻|=26
10^∗	(756, 151, 30)	11	11	11≡−1(mod𝑏); 𝑏=3,6,12	189	96	−11+27𝐻, |𝐻|=6
11^∗	(2380, 183, 14)	13	13	13^2≡−1(mod17)	35	15	13+5𝐻, |𝐻|=34
12^∗	(1056, 211, 42)	13	13	13^5≡−1(mod11)	1028	995	13+9𝐻, |𝐻|=22
13^∗	(1456, 195, 26)	13	13	13≡−1(mod7)	179	171	13+13𝐻, |𝐻|=14
14^∗	(1380, 197, 28)	13	13	13^2≡−1(mod𝑏); 𝑏=5,10	29	15	−13+21𝐻, |𝐻|=10
15^∗	(1548, 273, 48)	15	3, 5	𝑎^21≡−1(mod43); 𝑎=3,5	46	6	15+3𝐻, |𝐻|=86
16^∗	(1160, 305, 80)	15	3, 5	𝑎≡−1(mod29); 𝑎=3^14,5^7	49	33	15+5𝐻, |𝐻|=58
17^∗	(1012, 337, 112)	15	3, 5	5^11≡−1(mod23); Remark 2.3	13	5	15+7𝐻, |𝐻|=46
18^∗	(1300, 433, 144)	17	17	17^2≡−1(mod𝑏); 𝑏=5,10	50	16	−17+45𝐻, |𝐻|=10
19^∗	(5220, 307, 18)	17	17	17^2≡−1(mod𝑏); 𝑏=5,10	113	50	17+29𝐻, |𝐻|=10
20^∗	(5220, 307, 18)	17	17	17^2≡−1(mod29)	113	50	17+5𝐻, |𝐻|=58
21^∗	(5220, 307, 18)	17	17	17≡−1(mod𝑏); 𝑏=3,6	113	50	−17+27𝐻, |𝐻|=12
22^∗	(33216, 365, 4)	19	19	19^86≡−1(mod173)	?	?	19+𝐻, |𝐻|=346
23^∗	(11564, 373, 12)	19	19	19^3≡−1(mod7)	28	16	−19+28𝐻, |𝐻|=14
24^∗	(7240, 381, 20)	19	19	19^2≡−1(mod181)	?	?	19+𝐻, |𝐻|=362
25^∗	(4368, 397, 36)	19	19	19^3≡−1(mod7)	?	?	19+27𝐻, |𝐻|=14
26^∗	(4180, 399, 38)	19	19	19^5≡−1(mod11)	36	15	−19+19𝐻, |𝐻|=22
27^∗	(2508, 437, 76)	19	19	19^5≡−1(mod11)	34	20	19+19𝐻, |𝐻|=22
28^∗	(1976, 475, 114)	19	19	19^6≡−1(mod13)	39	30	−19+19𝐻, |𝐻|=26
29^∗	(1624, 541, 180)	19	19	19^14≡−1(mod29)	56	30	19+9𝐻, |𝐻|=58
30^∗	(1520, 589, 228)	19	19	19^3≡−1(mod𝑏); 𝑏=5,10	178	147	19+57𝐻, |𝐻|=10
31^∗	(97904, 443, 2)	21	3, 7	𝑎^105≡−1(mod211); 𝑎=3,7	?	?	21+𝐻, |𝐻|=422
32^∗	(11680, 459, 18)	21	3, 7	𝑎≡−1(mod73); 𝑎=3^6,7^12	?	?	21+3𝐻, |𝐻|=146
33^∗	(9724, 463, 22)	21	3, 7	𝑎^8≡−1(mod17); 𝑎=3,7	35	15	21+13𝐻, |𝐻|=34
34^∗	(7840, 469, 28)	21	3, 7	𝑎^2≡−1(mod5); 𝑎=3,7	?	?	−21+49𝐻, |𝐻|=10
35^∗	(7380, 471, 30)	21	3, 7	𝑎^2≡−1(mod5); 𝑎=3,7	149	66	21+45𝐻, |𝐻|=10
36^∗	(3128, 531, 90)	21	3, 7	𝑎^8≡−1(mod17); 𝑎=3,7	?	?	21+15𝐻, |𝐻|=34
37^∗	(2756, 551, 110)	21	3, 7	𝑎≡−1(mod53); 𝑎=3^26,7^13	20	5	21+5𝐻, |𝐻|=106
38^∗	(2296, 595, 154)	21	3, 7	𝑎≡−1(mod41); 𝑎=3^4,7^20	?	?	21+7𝐻, |𝐻|=82
39^∗	(1904, 693, 252)	21	3, 7	𝑎^8≡−1(mod17); 𝑎=3,7	186	147	−21+21𝐻, |𝐻|=34
40^∗	(1836, 735, 294)	21	3, 7	𝑎≡−1(mod17); 𝑎=3^2,7^2	117	56	21+21𝐻, |𝐻|=34
41^∗	(1820, 749, 308)	21	3, 7	𝑎≡−1(mod5); 𝑎=3^2,7^2	35	15	−21+77𝐻, |𝐻|=10
42^∗	(1800, 771, 330)	21	3, 7	𝑎≡−1(mod5); 𝑎=3^2,7^2	749	412	21+75𝐻, |𝐻|=10
43^∗	(47616, 535, 6)	23	23	23≡−1(mod3)	?	?	−23+93𝐻, |𝐻|=6
44^∗	(35980, 537, 8)	23	23	23^32≡−1(mod257)	35	15	23+𝐻, |𝐻|=514
45^∗	(12720, 553, 24)	23	23	23^2≡−1(mod5)	?	?	23+53𝐻, |𝐻|=10
46^∗	(12720, 553, 24)	23	23	23^2≡−1(mod53)	?	?	23+5𝐻,; |𝐻|=106
47^∗	(7176, 575, 46)	23	23	23^3≡−1(mod13)	?	?	−23+23𝐻, |𝐻|=26
48^∗	(4320, 617, 88)	23	23	23≡−1(mod3)	?	?	23+99𝐻, |𝐻|=6
49^∗	(3220, 667, 138)	23	23	23^2≡−1(mod5)	27	15	−23+69𝐻, |𝐻|=10
50^∗	(2760, 713, 184)	23	23	23≡−1(mod3)	?	?	23+115𝐻, |𝐻|=6
51^∗	(2760, 713, 184)	23	23	23^2≡−1(mod5)	?	?	23+69𝐻, |𝐻|=10
52^∗	(2380, 793, 264)	23	23	23^8≡−1(mod17)	35	15	−23+24𝐻, |𝐻|=34
53^∗	(2380, 793, 264)	23	23	23^2≡−1(mod5)	35	15	23+77𝐻, |𝐻|=10
54^∗	(196252, 627, 2)	25	5	5^3≡−1(mod7)	?	?	25+43𝐻, |𝐻|=14
55^∗	(196252, 627, 2)	25	5	5^21≡−1(mod43)	?	?	25+7𝐻, |𝐻|=86
56^∗	(40260, 635, 10)	25	5	5^15≡−1(mod61)	370	66	25+5𝐻, |𝐻|=122
57^∗	(16276, 651, 26)	25	5	5^4≡−1(mod313)	20	5	25+𝐻, |𝐻|=626
58^∗	(14280, 655, 30)	25	5	5^3≡−1(mod7)	?	?	25+45𝐻, |𝐻|=14
59^∗	(11040, 665, 40)	25	5	5≡−1(mod3)	?	?	−25+115𝐻, |𝐻|=6
60^∗	(9100, 675, 50)	25	5	5^2≡−1(mod13)	118	50	25+25𝐻, |𝐻|=26
61^∗	(6328, 703, 78)	25	5	5^56≡−1(mod113)	?	?	25+3𝐻, |𝐻|=226
62^∗	(4620, 745, 120)	25	5	5^3≡−1(mod7)	140	66	−25+55𝐻, |𝐻|=14
63^∗	(4380, 755, 130)	25	5	5^36≡−1(mod73)	53	15	25+5𝐻, |𝐻|=146
64^∗	(3400, 825, 200)	25	5	5^8≡−1(mod17)	?	?	−25+25𝐻, |𝐻|=34
65^∗	(3060, 875, 250)	25	5	5^8≡−1(mod17)	113	50	25+25𝐻, |𝐻|=34
66^∗	(2640, 1015, 390)	25	5	5≡−1(mod3)	?	?	25+165𝐻, |𝐻|=6
67^∗	(2520, 1145, 520)	25	5	5≡−1(mod3)	?	?	−25+195𝐻, |𝐻|=6