MinT - Best Known (105−59, 105, s)-Nets in Base 9

Best Known (105−59, 105, s)-Nets in Base 9

(105−59, 105, 94)-Net over F₉ — Constructive and digital

Digital (46, 105, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 33, 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, 72, 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]

(105−59, 105, 96)-Net in Base 9 — Constructive

(46, 105, 96)-net in base 9, using

base change [i] based on digital (11, 70, 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]

(105−59, 105, 162)-Net over F₉ — Digital

Digital (46, 105, 162)-net over F₉, using

net from sequence [i] based on digital (46, 161)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 46 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(105−59, 105, 3838)-Net in Base 9 — Upper bound on s

There is no (46, 105, 3839)-net in base 9, because

1 times m-reduction [i] would yield (46, 104, 3839)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1749 878570 878879 764005 792942 041304 969729 967827 573865 862540 681649 612178 882408 081323 376775 168220 878617 > 9¹⁰⁴ [i]