MinT - Best Known (137−81, 137, s)-Nets in Base 3

Best Known (137−81, 137, s)-Nets in Base 3

(137−81, 137, 48)-Net over F₃ — Constructive and digital

Digital (56, 137, 48)-net over F₃, using

t-expansion [i] based on digital (45, 137, 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]

(137−81, 137, 64)-Net over F₃ — Digital

Digital (56, 137, 64)-net over F₃, using

t-expansion [i] based on digital (49, 137, 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]

(137−81, 137, 292)-Net in Base 3 — Upper bound on s

There is no (56, 137, 293)-net in base 3, because

1 times m-reduction [i] would yield (56, 136, 293)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 80230 511640 029118 365648 420643 262249 619937 470481 149154 199897 502945 > 3¹³⁶ [i]