MinT - Best Known (99−59, 99, s)-Nets in Base 9

Best Known (99−59, 99, s)-Nets in Base 9

(99−59, 99, 81)-Net over F₉ — Constructive and digital

Digital (40, 99, 81)-net over F₉, using

t-expansion [i] based on digital (32, 99, 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]

(99−59, 99, 82)-Net in Base 9 — Constructive

(40, 99, 82)-net in base 9, using

base change [i] based on digital (7, 66, 82)-net over F₂₇, using
- net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
    - a function field by SÃ©mirat [i]

(99−59, 99, 140)-Net over F₉ — Digital

Digital (40, 99, 140)-net over F₉, using

t-expansion [i] based on digital (39, 99, 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]

(99−59, 99, 2429)-Net in Base 9 — Upper bound on s

There is no (40, 99, 2430)-net in base 9, because

1 times m-reduction [i] would yield (40, 98, 2430)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 3284 575664 908184 678875 726134 413308 621766 844895 770267 151453 728164 736063 398154 895971 360962 944945 > 9⁹⁸ [i]