MinT - Best Known (70−34, 70, s)-Nets in Base 3

Best Known (70−34, 70, s)-Nets in Base 3

(70−34, 70, 40)-Net over F₃ — Constructive and digital

Digital (36, 70, 40)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (4, 21, 12)-net over F₃, using
  - net from sequence [i] based on digital (4, 11)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [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, 49, 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]

(70−34, 70, 48)-Net over F₃ — Digital

Digital (36, 70, 48)-net over F₃, using

net from sequence [i] based on digital (36, 47)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 36 and N(F) ≥ 48, using
  - a curve from the manYPoints database [i]

(70−34, 70, 314)-Net in Base 3 — Upper bound on s

There is no (36, 70, 315)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2553 812963 442177 943142 880477 705431 > 3⁷⁰ [i]