MinT - Best Known (144−85, 144, s)-Nets in Base 9

Best Known (144−85, 144, s)-Nets in Base 9

(144−85, 144, 94)-Net over F₉ — Constructive and digital

Digital (59, 144, 94)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (4, 46, 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, 98, 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]

(144−85, 144, 96)-Net in Base 9 — Constructive

(59, 144, 96)-net in base 9, using

base change [i] based on digital (11, 96, 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]

(144−85, 144, 182)-Net over F₉ — Digital

Digital (59, 144, 182)-net over F₉, using

t-expansion [i] based on digital (50, 144, 182)-net over F₉, using
- net from sequence [i] based on digital (50, 181)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 50 and N(F) ≥ 182, using
    - a curve from the manYPoints database [i]

(144−85, 144, 3635)-Net in Base 9 — Upper bound on s

There is no (59, 144, 3636)-net in base 9, because

1 times m-reduction [i] would yield (59, 143, 3636)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 28626 109380 122304 049250 469160 985883 526816 627188 043576 673135 243061 856283 917610 369749 380999 067208 047831 237301 285012 619789 950864 619849 329345 > 9¹⁴³ [i]