Best Known (85, ∞, s)-Nets in Base 2
(85, ∞, 52)-Net over F2 — Constructive and digital
Digital (85, m, 52)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (85, 51)-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 3 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]
(85, ∞, 57)-Net over F2 — Digital
Digital (85, m, 57)-net over F2 for arbitrarily large m, using
- net from sequence [i] based on digital (85, 56)-sequence over F2, using
- t-expansion [i] based on digital (83, 56)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 83 and N(F) ≥ 57, using
- t-expansion [i] based on digital (83, 56)-sequence over F2, using
(85, ∞, 94)-Net in Base 2 — Upper bound on s
There is no (85, m, 95)-net in base 2 for arbitrarily large m, because
- m-reduction [i] would yield (85, 656, 95)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2656, 95, S2, 7, 571), but
- the (dual) Plotkin bound for OOAs shows that M ≥ 48 139661 438954 597921 667719 398539 061055 180622 375617 701401 280511 697340 550474 668248 873530 291822 401735 580998 789486 434628 375479 640487 371829 773449 211312 372139 719675 646518 399598 766126 749630 133154 821435 293696 / 143 > 2656 [i]
- extracting embedded OOA [i] would yield OOA(2656, 95, S2, 7, 571), but