MinT - Best Known (45, 45+61, s)-Nets in Base 9

Best Known (45, 45+61, s)-Nets in Base 9

(45, 45+61, 84)-Net over F₉ — Constructive and digital

Digital (45, 106, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 32, 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, 74, 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]

(45, 45+61, 88)-Net in Base 9 — Constructive

(45, 106, 88)-net in base 9, using

2 times m-reduction [i] based on (45, 108, 88)-net in base 9, using
- base change [i] based on digital (9, 72, 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]

(45, 45+61, 147)-Net over F₉ — Digital

Digital (45, 106, 147)-net over F₉, using

t-expansion [i] based on digital (43, 106, 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]

(45, 45+61, 3274)-Net in Base 9 — Upper bound on s

There is no (45, 106, 3275)-net in base 9, because

1 times m-reduction [i] would yield (45, 105, 3275)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 15763 851006 468176 120937 865563 483985 583371 985430 106247 018377 579421 256656 550849 709825 560333 920532 914129 > 9¹⁰⁵ [i]