MinT - Best Known (112−71, 112, s)-Nets in Base 9

Best Known (112−71, 112, s)-Nets in Base 9

(112−71, 112, 81)-Net over F₉ — Constructive and digital

Digital (41, 112, 81)-net over F₉, using

t-expansion [i] based on digital (32, 112, 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]

(112−71, 112, 140)-Net over F₉ — Digital

Digital (41, 112, 140)-net over F₉, using

t-expansion [i] based on digital (39, 112, 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]

(112−71, 112, 1825)-Net in Base 9 — Upper bound on s

There is no (41, 112, 1826)-net in base 9, because

1 times m-reduction [i] would yield (41, 111, 1826)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 8370 414499 029271 888263 434074 032502 763199 693823 271081 937482 780685 660158 094318 917226 911159 114953 377038 298993 > 9¹¹¹ [i]