MinT - Best Known (116−75, 116, s)-Nets in Base 9

Best Known (116−75, 116, s)-Nets in Base 9

(116−75, 116, 81)-Net over F₉ — Constructive and digital

Digital (41, 116, 81)-net over F₉, using

t-expansion [i] based on digital (32, 116, 81)-net over F₉, using
- net from sequence [i] based on digital (32, 80)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 32 and N(F) ≥ 81, using
    - F₄ from the tower of function fields by Bezerra and GarcÃa over F₉ [i]

(116−75, 116, 140)-Net over F₉ — Digital

Digital (41, 116, 140)-net over F₉, using

t-expansion [i] based on digital (39, 116, 140)-net over F₉, using
- net from sequence [i] based on digital (39, 139)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 39 and N(F) ≥ 140, using
    - a curve from the manYPoints database [i]

(116−75, 116, 1670)-Net in Base 9 — Upper bound on s

There is no (41, 116, 1671)-net in base 9, because

1 times m-reduction [i] would yield (41, 115, 1671)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 55 079693 694453 407040 123899 095776 970167 564471 030977 471600 261542 620364 716841 059101 387936 890861 269918 520560 865817 > 9¹¹⁵ [i]