MinT - Best Known (135−75, 135, s)-Nets in Base 3

Best Known (135−75, 135, s)-Nets in Base 3

(135−75, 135, 48)-Net over F₃ — Constructive and digital

Digital (60, 135, 48)-net over F₃, using

t-expansion [i] based on digital (45, 135, 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]

(135−75, 135, 64)-Net over F₃ — Digital

Digital (60, 135, 64)-net over F₃, using

t-expansion [i] based on digital (49, 135, 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]

(135−75, 135, 356)-Net in Base 3 — Upper bound on s

There is no (60, 135, 357)-net in base 3, because

1 times m-reduction [i] would yield (60, 134, 357)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 9059 901900 644015 754148 565232 207350 454233 964526 452354 613537 199451 > 3¹³⁴ [i]