Best Known (123, 123+∞, s)-Nets in Base 2
(123, 123+∞, 57)-Net over F2 — Constructive and digital
Digital (123, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (123, 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
(123, 123+∞, 80)-Net over F2 — Digital
Digital (123, m, 80)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (123, 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
(123, 123+∞, 133)-Net in Base 2 — Upper bound on s
There is no (123, m, 134)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (123, 928, 134)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2928, 134, S2, 7, 805), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 1 107275 774135 067954 476096 342616 503102 636802 640395 789331 084009 807351 957764 731258 139480 655116 995428 923168 673366 103537 247998 995723 876687 490822 451668 279385 328029 452651 626798 035306 524705 456541 652367 542536 510531 291963 739869 783583 002327 702363 074162 124855 262008 480852 449985 091336 055740 694528 / 403 > 2928 [i]
- extracting embedded OOA [i] would yield OOA(2928, 134, S2, 7, 805), but