MinT - Best Known (140−79, 140, s)-Nets in Base 3

Best Known (140−79, 140, s)-Nets in Base 3

(140−79, 140, 48)-Net over F₃ — Constructive and digital

Digital (61, 140, 48)-net over F₃, using

t-expansion [i] based on digital (45, 140, 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]

(140−79, 140, 64)-Net over F₃ — Digital

Digital (61, 140, 64)-net over F₃, using

t-expansion [i] based on digital (49, 140, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(140−79, 140, 349)-Net in Base 3 — Upper bound on s

There is no (61, 140, 350)-net in base 3, because

1 times m-reduction [i] would yield (61, 139, 350)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 241822 412273 303356 391266 503864 643338 094566 977247 204261 587284 631929 > 3¹³⁹ [i]