Best Known (133, 133+∞, s)-Nets in Base 2
(133, 133+∞, 57)-Net over F2 — Constructive and digital
Digital (133, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (133, 56)-sequence over F2, using
- t-expansion [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]
- t-expansion [i] based on digital (110, 56)-sequence over F2, using
(133, 133+∞, 81)-Net over F2 — Digital
Digital (133, m, 81)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (133, 80)-sequence over F2, using
- t-expansion [i] based on digital (126, 80)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 126 and N(F) ≥ 81, using
- t-expansion [i] based on digital (126, 80)-sequence over F2, using
(133, 133+∞, 143)-Net in Base 2 — Upper bound on s
There is no (133, m, 144)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (133, 998, 144)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2998, 144, S2, 7, 865), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1200 089640 048619 399462 236054 947202 027828 773389 110197 640337 000434 974793 177259 928457 192382 184273 579361 102906 033667 659524 444209 602719 975391 728830 272688 366240 378040 671590 780313 870367 160355 766985 567890 330868 401141 205198 161177 421948 661966 626540 921960 631046 431974 005500 892722 777968 141145 093931 325767 034823 770112 / 433 > 2998 [i]
- extracting embedded OOA [i] would yield OOA(2998, 144, S2, 7, 865), but