Best Known (124, ∞, s)-Nets in Base 2
(124, ∞, 57)-Net over F2 — Constructive and digital
Digital (124, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (124, 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
(124, ∞, 80)-Net over F2 — Digital
Digital (124, m, 80)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (124, 79)-sequence over F2, using
- t-expansion [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
- t-expansion [i] based on digital (121, 79)-sequence over F2, using
(124, ∞, 134)-Net in Base 2 — Upper bound on s
There is no (124, m, 135)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (124, 935, 135)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2935, 135, S2, 7, 811), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 70 284783 564770 215077 564672 764116 393662 454095 469057 316228 807966 787980 007623 597237 968345 846114 857390 008345 955304 145839 741837 892505 746458 433844 800976 684917 870984 273231 130524 470604 322287 339824 228969 257399 488150 204649 193045 934973 524801 041800 379602 417042 204866 194437 481020 879560 784065 069056 / 203 > 2935 [i]
- extracting embedded OOA [i] would yield OOA(2935, 135, S2, 7, 811), but