MinT - Best Known (143−85, 143, s)-Nets in Base 9

Best Known (143−85, 143, s)-Nets in Base 9

(143−85, 143, 92)-Net over F₉ — Constructive and digital

Digital (58, 143, 92)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 45, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [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]

(143−85, 143, 94)-Net in Base 9 — Constructive

(58, 143, 94)-net in base 9, using

1 times m-reduction [i] based on (58, 144, 94)-net in base 9, using
- base change [i] based on digital (10, 96, 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]

(143−85, 143, 182)-Net over F₉ — Digital

Digital (58, 143, 182)-net over F₉, using

t-expansion [i] based on digital (50, 143, 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]

(143−85, 143, 3449)-Net in Base 9 — Upper bound on s

There is no (58, 143, 3450)-net in base 9, because

1 times m-reduction [i] would yield (58, 142, 3450)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3205 137753 357592 822295 908865 627745 882662 320102 169895 684577 072221 216160 135970 350246 814449 502742 356296 892453 996340 658483 765259 944924 585185 > 9¹⁴² [i]