Best Known (106, ∞, s)-Nets in Base 2
(106, ∞, 56)-Net over F2 — Constructive and digital
Digital (106, m, 56)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (106, 55)-sequence over F2, using
- t-expansion [i] based on digital (105, 55)-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 7 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 (105, 55)-sequence over F2, using
(106, ∞, 65)-Net over F2 — Digital
Digital (106, m, 65)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (106, 64)-sequence over F2, using
- t-expansion [i] based on digital (95, 64)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 95 and N(F) ≥ 65, using
- t-expansion [i] based on digital (95, 64)-sequence over F2, using
(106, ∞, 116)-Net in Base 2 — Upper bound on s
There is no (106, m, 117)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (106, 693, 117)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2693, 117, S2, 6, 587), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 2 054740 586542 333401 266011 673000 050259 980601 485477 802288 866515 977761 223497 772297 196138 150990 733432 988760 540222 209444 616294 148215 722728 048384 034302 644785 890957 013759 246534 548671 168618 655423 563561 434098 926459 289600 / 49 > 2693 [i]
- extracting embedded OOA [i] would yield OOA(2693, 117, S2, 6, 587), but