MinT - Best Known (100−57, 100, s)-Nets in Base 9

Best Known (100−57, 100, s)-Nets in Base 9

(100−57, 100, 84)-Net over F₉ — Constructive and digital

Digital (43, 100, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 30, 20)-net over F₉, using
  - net from sequence [i] based on digital (2, 19)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 2 and N(F) ≥ 20, using
      - a function field by GarcÃa and Quoos [i]
2. digital (13, 70, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(100−57, 100, 88)-Net in Base 9 — Constructive

(43, 100, 88)-net in base 9, using

2 times m-reduction [i] based on (43, 102, 88)-net in base 9, using
- base change [i] based on digital (9, 68, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(100−57, 100, 147)-Net over F₉ — Digital

Digital (43, 100, 147)-net over F₉, using

net from sequence [i] based on digital (43, 146)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 43 and N(F) ≥ 147, using
  - a curve from the manYPoints database [i]

(100−57, 100, 3323)-Net in Base 9 — Upper bound on s

There is no (43, 100, 3324)-net in base 9, because

1 times m-reduction [i] would yield (43, 99, 3324)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 29568 334774 518563 465766 723339 686745 708390 258607 561621 804144 239610 299074 899079 193007 611274 273665 > 9⁹⁹ [i]