MinT - Best Known (78−41, 78, s)-Nets in Base 9

Best Known (78−41, 78, s)-Nets in Base 9

(78−41, 78, 94)-Net over F₉ — Constructive and digital

Digital (37, 78, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 24, 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, 54, 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]

(78−41, 78, 96)-Net in Base 9 — Constructive

(37, 78, 96)-net in base 9, using

base change [i] based on digital (11, 52, 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]

(78−41, 78, 132)-Net over F₉ — Digital

Digital (37, 78, 132)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(78−41, 78, 4886)-Net in Base 9 — Upper bound on s

There is no (37, 78, 4887)-net in base 9, because

1 times m-reduction [i] would yield (37, 77, 4887)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 30 047492 837500 253882 979984 825697 893421 568103 984848 942658 248974 680313 242721 > 9⁷⁷ [i]