| v | k | lambda | n | Status | Comment |
| 1023 | 511 | 255 | 256 | EXISTS | Complete list determined by Gaal and Golomb, Math. Comp., vol 70 (2000), pp. 357-366. |
| 3439 | 1719 | 859 | 860 | OPEN | cyclic Hadamard difference set |
| 4355 | 2177 | 1088 | 1089 | OPEN | cyclic Hadamard difference set |
| 4623 | 2311 | 1155 | 1156 | NO | Thm. 4.19 of Lander's book |
| 5775 | 2887 | 1443 | 1444 | NO | Thm. 4.19 of Lander's book |
| 7395 | 3697 | 1848 | 1849 | NO | Thm. 4.20 of Lander's book |
| 7743 | 3871 | 1935 | 1936 | NO | Thm. 4.19 of Lander's book |
| 8227 | 4113 | 2056 | 2057 | NO | Thm. 4.20 of Lander's book |
| 8463 | 4231 | 2115 | 2116 | NO | Thm. 4.19 of Lander's book |
| 8591 | 4295 | 2147 | 2148 | OPEN | cyclic Hadamard difference set |
| 8835 | 4417 | 2208 | 2209 | OPEN | cyclic Hadamard difference set |
| 9135 | 4567 | 2283 | 2284 | OPEN | cyclic Hadamard difference set |
| 9215 | 4607 | 2303 | 2304 | OPEN | cyclic Hadamard difference set |
| 9423 | 4711 | 2355 | 2356 | OPEN | cyclic Hadamard difference set |