Best Known (212−99, 212, s)-Nets in Base 2
(212−99, 212, 57)-Net over F2 — Constructive and digital
Digital (113, 212, 57)-net over F2, using
- t-expansion [i] based on digital (110, 212, 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
(212−99, 212, 72)-Net over F2 — Digital
Digital (113, 212, 72)-net over F2, using
- t-expansion [i] based on digital (110, 212, 72)-net over F2, using
- net from sequence [i] based on digital (110, 71)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 110 and N(F) ≥ 72, using
- net from sequence [i] based on digital (110, 71)-sequence over F2, using
(212−99, 212, 287)-Net in Base 2 — Upper bound on s
There is no (113, 212, 288)-net in base 2, because
- 1 times m-reduction [i] would yield (113, 211, 288)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2211, 288, S2, 98), but
- 12 times code embedding in larger space [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]
- 12 times code embedding in larger space [i] would yield OA(2223, 300, S2, 98), but
- extracting embedded orthogonal array [i] would yield OA(2211, 288, S2, 98), but