MinT - Best Known (42, 42+55, s)-Nets in Base 9

Best Known (42, 42+55, s)-Nets in Base 9

(42, 42+55, 84)-Net over F₉ — Constructive and digital

Digital (42, 97, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 29, 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, 68, 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]

(42, 42+55, 88)-Net in Base 9 — Constructive

(42, 97, 88)-net in base 9, using

2 times m-reduction [i] based on (42, 99, 88)-net in base 9, using
- base change [i] based on digital (9, 66, 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]

(42, 42+55, 140)-Net over F₉ — Digital

Digital (42, 97, 140)-net over F₉, using

t-expansion [i] based on digital (39, 97, 140)-net over F₉, using
- net from sequence [i] based on digital (39, 139)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
    - a curve from the manYPoints database [i]

(42, 42+55, 3357)-Net in Base 9 — Upper bound on s

There is no (42, 97, 3358)-net in base 9, because

1 times m-reduction [i] would yield (42, 96, 3358)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 40 512146 460184 438077 803951 615047 931690 885936 978982 775968 663677 606729 506524 436184 253665 776913 > 9⁹⁶ [i]