MinT - Best Known (72−35, 72, s)-Nets in Base 3

Best Known (72−35, 72, s)-Nets in Base 3

(72−35, 72, 41)-Net over F₃ — Constructive and digital

Digital (37, 72, 41)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (5, 22, 13)-net over F₃, using
  - net from sequence [i] based on digital (5, 12)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 4, N(F)Â =Â 12, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 4 and N(F) ≥ 12, using an explicitly constructive algebraic function field [i]
2. digital (15, 50, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]

(72−35, 72, 52)-Net over F₃ — Digital

Digital (37, 72, 52)-net over F₃, using

net from sequence [i] based on digital (37, 51)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 37 and N(F) ≥ 52, using
  - a curve from the manYPoints database [i]

(72−35, 72, 336)-Net in Base 3 — Upper bound on s

There is no (37, 72, 337)-net in base 3, because

1 times m-reduction [i] would yield (37, 71, 337)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7629 235924 535281 232975 565671 542371 > 3⁷¹ [i]