MinT - Best Known (142−89, 142, s)-Nets in Base 9

Best Known (142−89, 142, s)-Nets in Base 9

(142−89, 142, 81)-Net over F₉ — Constructive and digital

Digital (53, 142, 81)-net over F₉, using

t-expansion [i] based on digital (32, 142, 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]

(142−89, 142, 182)-Net over F₉ — Digital

Digital (53, 142, 182)-net over F₉, using

t-expansion [i] based on digital (50, 142, 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]

(142−89, 142, 2437)-Net in Base 9 — Upper bound on s

There is no (53, 142, 2438)-net in base 9, because

1 times m-reduction [i] would yield (53, 141, 2438)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 355 333132 087975 557065 722831 525177 839577 896749 099326 114337 931416 301440 936895 749402 086626 323219 683545 987215 179414 728403 467459 830675 390145 > 9¹⁴¹ [i]