MinT - Best Known (19, 19+∞, s)-Nets in Base 3

Best Known (19, 19+∞, s)-Nets in Base 3

(19, 19+∞, 28)-Net over F₃ — Constructive and digital

Digital (19, m, 28)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (19, 27)-sequence over F₃, using
- t-expansion [i] based on digital (15, 27)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
    - an explicitly constructive algebraic function field [i]

(19, 19+∞, 32)-Net over F₃ — Digital

Digital (19, m, 32)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (19, 31)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 19 and N(F) ≥ 32, using
  - a curve from the manYPoints database [i]

(19, 19+∞, 47)-Net in Base 3 — Upper bound on s

There is no (19, m, 48)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (19, 187, 48)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3¹⁸⁷, 48, S₃, 4, 168), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 28 488576 090148 975807 870554 503772 645936 738615 954063 548485 251933 552601 835352 135466 459456 323177 / 169 > 3¹⁸⁷ [i]