MinT - Best Known (85−51, 85, s)-Nets in Base 3

Best Known (85−51, 85, s)-Nets in Base 3

(85−51, 85, 38)-Net over F₃ — Constructive and digital

Digital (34, 85, 38)-net over F₃, using

t-expansion [i] based on digital (32, 85, 38)-net over F₃, using
- net from sequence [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]

(85−51, 85, 46)-Net over F₃ — Digital

Digital (34, 85, 46)-net over F₃, using

t-expansion [i] based on digital (33, 85, 46)-net over F₃, using
- net from sequence [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]

(85−51, 85, 180)-Net in Base 3 — Upper bound on s

There is no (34, 85, 181)-net in base 3, because

1 times m-reduction [i] would yield (34, 84, 181)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12423 851030 435601 754145 611470 138197 451147 > 3⁸⁴ [i]