MinT - Best Known (135−75, 135, s)-Nets in Base 9

Best Known (135−75, 135, s)-Nets in Base 9

(135−75, 135, 108)-Net over F₉ — Constructive and digital

Digital (60, 135, 108)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (6, 43, 34)-net over F₉, using
  - net from sequence [i] based on digital (6, 33)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 6 and N(F) ≥ 34, using
      - a function field by SÃ©mirat [i]
2. digital (17, 92, 74)-net over F₉, using
  - net from sequence [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(135−75, 135, 190)-Net over F₉ — Digital

Digital (60, 135, 190)-net over F₉, using

net from sequence [i] based on digital (60, 189)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 60 and N(F) ≥ 190, using
  - Drinfeld modules of rank 1 [i]

(135−75, 135, 5210)-Net in Base 9 — Upper bound on s

There is no (60, 135, 5211)-net in base 9, because

1 times m-reduction [i] would yield (60, 134, 5211)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 74 336252 534086 687731 425647 994581 714718 547560 589166 778000 187077 941333 077675 227029 330149 441476 109159 115493 441868 000947 869510 851129 > 9¹³⁴ [i]