MinT - Best Known (135−89, 135, s)-Nets in Base 9

Best Known (135−89, 135, s)-Nets in Base 9

(135−89, 135, 81)-Net over F₉ — Constructive and digital

Digital (46, 135, 81)-net over F₉, using

t-expansion [i] based on digital (32, 135, 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]

(135−89, 135, 162)-Net over F₉ — Digital

Digital (46, 135, 162)-net over F₉, using

net from sequence [i] based on digital (46, 161)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 46 and N(F) ≥ 162, using
  - a curve from the manYPoints database [i]

(135−89, 135, 1710)-Net in Base 9 — Upper bound on s

There is no (46, 135, 1711)-net in base 9, because

1 times m-reduction [i] would yield (46, 134, 1711)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 74 508643 325315 967313 570154 309745 701994 175892 130252 141773 633164 611092 213928 067896 807204 275548 088955 234446 818589 919313 267201 352865 > 9¹³⁴ [i]