Best Known (247−123, 247, s)-Nets in Base 2
(247−123, 247, 57)-Net over F2 — Constructive and digital
Digital (124, 247, 57)-net over F2, using
- t-expansion [i] based on digital (110, 247, 57)-net over F2, using
- net from sequence [i] based on digital (110, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 8 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (110, 56)-sequence over F2, using
(247−123, 247, 80)-Net over F2 — Digital
Digital (124, 247, 80)-net over F2, using
- t-expansion [i] based on digital (121, 247, 80)-net over F2, using
- net from sequence [i] based on digital (121, 79)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 121 and N(F) ≥ 80, using
- net from sequence [i] based on digital (121, 79)-sequence over F2, using
(247−123, 247, 263)-Net over F2 — Upper bound on s (digital)
There is no digital (124, 247, 264)-net over F2, because
- 1 times m-reduction [i] would yield digital (124, 246, 264)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2246, 264, F2, 122) (dual of [264, 18, 123]-code), but
- residual code [i] would yield OA(2124, 141, S2, 61), but
- 1 times truncation [i] would yield OA(2123, 140, S2, 60), but
- the linear programming bound shows that M ≥ 3 888746 889172 484760 459445 013730 247120 519168 / 329189 > 2123 [i]
- 1 times truncation [i] would yield OA(2123, 140, S2, 60), but
- residual code [i] would yield OA(2124, 141, S2, 61), but
- extracting embedded orthogonal array [i] would yield linear OA(2246, 264, F2, 122) (dual of [264, 18, 123]-code), but
(247−123, 247, 267)-Net in Base 2 — Upper bound on s
There is no (124, 247, 268)-net in base 2, because
- 15 times m-reduction [i] would yield (124, 232, 268)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2232, 268, S2, 108), but
- the linear programming bound shows that M ≥ 31 015917 364418 637531 270768 022329 527868 589587 272176 065660 880357 772788 956541 131711 053824 / 4226 244518 828475 > 2232 [i]
- extracting embedded orthogonal array [i] would yield OA(2232, 268, S2, 108), but