MinT - Best Known (70−40, 70, s)-Nets in Base 16

Best Known (70−40, 70, s)-Nets in Base 16

(70−40, 70, 110)-Net over F₁₆ — Constructive and digital

Digital (30, 70, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 24, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 46, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(70−40, 70, 129)-Net in Base 16 — Constructive

(30, 70, 129)-net in base 16, using

base change [i] based on digital (0, 40, 129)-net over F₁₂₈, using
- net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
  - generalized Faure sequence [i]
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
    - the rational function field F₁₂₈(x) [i]
  - Niederreiter sequence [i]

(70−40, 70, 162)-Net over F₁₆ — Digital

Digital (30, 70, 162)-net over F₁₆, using

net from sequence [i] based on digital (30, 161)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 30 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(70−40, 70, 9059)-Net in Base 16 — Upper bound on s

There is no (30, 70, 9060)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 943753 241280 487864 503524 269491 326757 016683 413155 318378 561415 111698 669881 458804 522376 > 16⁷⁰ [i]