MinT - Best Known (34, ∞, s)-Nets in Base 3

Best Known (34, ∞, s)-Nets in Base 3

(34, ∞, 38)-Net over F₃ — Constructive and digital

Digital (34, m, 38)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (34, 37)-sequence over F₃, using
- t-expansion [i] based on digital (32, 37)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
    - an explicitly constructive algebraic function field [i]

(34, ∞, 46)-Net over F₃ — Digital

Digital (34, m, 46)-net over F₃ for arbitrarily large m, using

net from sequence [i] based on digital (34, 45)-sequence over F₃, using
- t-expansion [i] based on digital (33, 45)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 33 and N(F) ≥ 46, using
    - a curve from the manYPoints database [i]

(34, ∞, 79)-Net in Base 3 — Upper bound on s

There is no (34, m, 80)-net in base 3 for arbitrarily large m, because

m-reduction [i] would yield (34, 314, 80)-net in base 3, but
- extracting embedded OOA [i] would yield OOA(3³¹⁴, 80, S₃, 4, 280), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 241 601901 558808 255480 810106 750959 761525 496530 550725 667504 364320 172551 252618 533199 343528 859969 183401 331740 222554 999779 831546 868843 129811 039523 483186 441561 / 281 > 3³¹⁴ [i]