MinT - Best Known (212−166, 212, s)-Nets in Base 3

Best Known (212−166, 212, s)-Nets in Base 3

(212−166, 212, 48)-Net over F₃ — Constructive and digital

Digital (46, 212, 48)-net over F₃, using

t-expansion [i] based on digital (45, 212, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(212−166, 212, 56)-Net over F₃ — Digital

Digital (46, 212, 56)-net over F₃, using

t-expansion [i] based on digital (40, 212, 56)-net over F₃, 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]

(212−166, 212, 145)-Net in Base 3 — Upper bound on s

There is no (46, 212, 146)-net in base 3, because

79 times m-reduction [i] would yield (46, 133, 146)-net in base 3, but
- extracting embedded orthogonal array [i] would yield OA(3¹³³, 146, S₃, 87), but
  - the linear programming bound shows that MÂ â‰¥ 1497 523208 240534 920642 237582 447721 349397 399972 787749 000691 443934 141439 / 508571 > 3¹³³ [i]