Best Known (126, 126+99, s)-Nets in Base 2
(126, 126+99, 63)-Net over F2 — Constructive and digital
Digital (126, 225, 63)-net over F2, using
- 3 times m-reduction [i] based on digital (126, 228, 63)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (21, 72, 21)-net over F2, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 21 and N(F) ≥ 21, using
- net from sequence [i] based on digital (21, 20)-sequence over F2, using
- digital (54, 156, 42)-net over F2, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 54 and N(F) ≥ 42, using
- net from sequence [i] based on digital (54, 41)-sequence over F2, using
- digital (21, 72, 21)-net over F2, using
- (u, u+v)-construction [i] based on
(126, 126+99, 82)-Net over F2 — Digital
Digital (126, 225, 82)-net over F2, using
(126, 126+99, 347)-Net over F2 — Upper bound on s (digital)
There is no digital (126, 225, 348)-net over F2, because
- 1 times m-reduction [i] would yield digital (126, 224, 348)-net over F2, but
- extracting embedded orthogonal array [i] would yield linear OA(2224, 348, F2, 98) (dual of [348, 124, 99]-code), but
- construction Y1 [i] would yield
- OA(2223, 300, S2, 98), but
- the linear programming bound shows that M ≥ 1 797227 776840 883681 896516 291018 079602 556004 736527 384481 376471 307374 972535 986485 205069 933364 379648 / 112622 624143 760877 566730 935625 > 2223 [i]
- linear OA(2124, 348, F2, 48) (dual of [348, 224, 49]-code), but
- discarding factors / shortening the dual code would yield linear OA(2124, 332, F2, 48) (dual of [332, 208, 49]-code), but
- the improved Johnson bound shows that N ≤ 1985 666223 770950 387579 854572 849250 190195 414082 522040 623254 891664 < 2208 [i]
- discarding factors / shortening the dual code would yield linear OA(2124, 332, F2, 48) (dual of [332, 208, 49]-code), but
- OA(2223, 300, S2, 98), but
- construction Y1 [i] would yield
- extracting embedded orthogonal array [i] would yield linear OA(2224, 348, F2, 98) (dual of [348, 124, 99]-code), but
(126, 126+99, 386)-Net in Base 2 — Upper bound on s
There is no (126, 225, 387)-net in base 2, because
- 1 times m-reduction [i] would yield (126, 224, 387)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 28 960408 360049 549862 847478 604306 577035 938216 969759 521334 410136 192224 > 2224 [i]