MinT - Best Known (70−45, 70, s)-Nets in Base 128

Best Known (70−45, 70, s)-Nets in Base 128

(70−45, 70, 321)-Net over F₁₂₈ — Constructive and digital

Digital (25, 70, 321)-net over F₁₂₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 22, 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]
2. digital (3, 48, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(70−45, 70, 513)-Net over F₁₂₈ — Digital

Digital (25, 70, 513)-net over F₁₂₈, using

t-expansion [i] based on digital (24, 70, 513)-net over F₁₂₈, using
- net from sequence [i] based on digital (24, 512)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 24 and N(F) ≥ 513, using
    - a curve from the manYPoints database [i]

(70−45, 70, 289741)-Net in Base 128 — Upper bound on s

There is no (25, 70, 289742)-net in base 128, because

1 times m-reduction [i] would yield (25, 69, 289742)-net in base 128, but
- the generalized Rao bound for nets shows that 128^mÂ â‰¥ 24 975787 872254 247742 420389 346286 591849 125570 485902 470326 575662 375959 543254 514695 447769 971202 466171 824136 908401 527999 375018 388041 378828 120950 925008 > 128⁶⁹ [i]