MinT - Best Known (78−42, 78, s)-Nets in Base 9

Best Known (78−42, 78, s)-Nets in Base 9

(78−42, 78, 84)-Net over F₉ — Constructive and digital

Digital (36, 78, 84)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (2, 23, 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, 55, 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]

(78−42, 78, 94)-Net in Base 9 — Constructive

(36, 78, 94)-net in base 9, using

base change [i] based on digital (10, 52, 94)-net over F₂₇, using
- net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
    - a function field by SÃ©mirat [i]

(78−42, 78, 128)-Net over F₉ — Digital

Digital (36, 78, 128)-net over F₉, using

t-expansion [i] based on digital (33, 78, 128)-net over F₉, using
- net from sequence [i] based on digital (33, 127)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 33 and N(F) ≥ 128, using
    - a curve from the manYPoints database [i]

(78−42, 78, 3786)-Net in Base 9 — Upper bound on s

There is no (36, 78, 3787)-net in base 9, because

the generalized Rao bound for nets shows that 9^mÂ â‰¥ 269 818102 238500 775201 725662 290732 719075 222603 891546 465036 385367 556924 823097 > 9⁷⁸ [i]