MinT - Best Known (89−57, 89, s)-Nets in Base 9

Best Known (89−57, 89, s)-Nets in Base 9

(89−57, 89, 81)-Net over F₉ — Constructive and digital

Digital (32, 89, 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]

(89−57, 89, 120)-Net over F₉ — Digital

Digital (32, 89, 120)-net over F₉, using

t-expansion [i] based on digital (31, 89, 120)-net over F₉, using
- net from sequence [i] based on digital (31, 119)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 31 and N(F) ≥ 120, using
    - a curve from the manYPoints database [i]

(89−57, 89, 1392)-Net in Base 9 — Upper bound on s

There is no (32, 89, 1393)-net in base 9, because

1 times m-reduction [i] would yield (32, 88, 1393)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 954974 387588 584331 158643 627075 214465 297213 605366 238867 704049 567420 549898 873276 951905 > 9⁸⁸ [i]