MinT - Best Known (41, ∞, s)-Nets in Base 3

Best Known (41, ∞, s)-Nets in Base 3

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

Digital (41, m, 42)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (41, 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]

(41, ∞, 56)-Net over F₃ — Digital

Digital (41, m, 56)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (41, 55)-sequence over F₃, using
- t-expansion [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]

(41, ∞, 93)-Net in Base 3 — Upper bound on s

There is no (41, m, 94)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (41, 371, 94)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3³⁷¹, 94, S₃, 4, 330), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 354653 782470 898582 729208 760309 092060 019323 578542 789858 618462 492562 367387 864732 444902 243350 628745 646801 486471 706952 149392 189258 292134 783728 500628 270340 405073 223308 595605 657211 543715 / 331 > 3³⁷¹ [i]