MinT - Best Known (40, ∞, s)-Nets in Base 3

Best Known (40, ∞, s)-Nets in Base 3

(40, ∞, 42)-Net over F₃ — Constructive and digital

Digital (40, m, 42)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (40, 41)-sequence over F₃, using
- t-expansion [i] based on digital (39, 41)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 39 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(40, ∞, 56)-Net over F₃ — Digital

Digital (40, m, 56)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (40, 55)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 40 and N(F) ≥ 56, using
  - a curve from the manYPoints database [i]

(40, ∞, 91)-Net in Base 3 — Upper bound on s

There is no (40, m, 92)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (40, 363, 92)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3³⁶³, 92, S₃, 4, 323), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 168286 662303 730863 643615 296801 545555 681715 967451 921423 155000 514749 907683 286978 494846 457366 460224 244406 583604 225224 434212 599631 542753 099233 585309 765906 718871 299729 525745 776829 > 3³⁶³ [i]