MinT - Best Known (50, 50+75, s)-Nets in Base 9

Best Known (50, 50+75, s)-Nets in Base 9

(50, 50+75, 81)-Net over F₉ — Constructive and digital

Digital (50, 125, 81)-net over F₉, using

t-expansion [i] based on digital (32, 125, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(50, 50+75, 84)-Net in Base 9 — Constructive

(50, 125, 84)-net in base 9, using

1 times m-reduction [i] based on (50, 126, 84)-net in base 9, using
- base change [i] based on digital (8, 84, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(50, 50+75, 182)-Net over F₉ — Digital

Digital (50, 125, 182)-net over F₉, using

net from sequence [i] based on digital (50, 181)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
  - a curve from the manYPoints database [i]

(50, 50+75, 2866)-Net in Base 9 — Upper bound on s

There is no (50, 125, 2867)-net in base 9, because

1 times m-reduction [i] would yield (50, 124, 2867)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 21205 015557 877241 792916 998194 765199 525231 985466 105090 628331 412413 251044 230212 406466 181021 187861 514283 519016 523573 130745 > 9¹²⁴ [i]