MinT - Best Known (96−53, 96, s)-Nets in Base 9

Best Known (96−53, 96, s)-Nets in Base 9

(96−53, 96, 94)-Net over F₉ — Constructive and digital

Digital (43, 96, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 30, 30)-net over F₉, using
  - net from sequence [i] based on digital (4, 29)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 4 and N(F) ≥ 30, using
      - a function field by SÃ©mirat [i]
2. digital (13, 66, 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]

(96−53, 96, 96)-Net in Base 9 — Constructive

(43, 96, 96)-net in base 9, using

base change [i] based on digital (11, 64, 96)-net over F₂₇, using
- net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
    - an explicitly constructive algebraic function field [i]

(96−53, 96, 147)-Net over F₉ — Digital

Digital (43, 96, 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]

(96−53, 96, 4028)-Net in Base 9 — Upper bound on s

There is no (43, 96, 4029)-net in base 9, because

1 times m-reduction [i] would yield (43, 95, 4029)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 506646 899207 318004 807107 580742 239100 272117 537989 987431 811369 378928 609485 237424 529213 502801 > 9⁹⁵ [i]