MinT - Best Known (87−45, 87, s)-Nets in Base 9

Best Known (87−45, 87, s)-Nets in Base 9

(87−45, 87, 102)-Net over F₉ — Constructive and digital

Digital (42, 87, 102)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 25, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]
2. digital (17, 62, 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]

(87−45, 87, 154)-Net over F₉ — Digital

Digital (42, 87, 154)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−45, 87, 6067)-Net in Base 9 — Upper bound on s

There is no (42, 87, 6068)-net in base 9, because

1 times m-reduction [i] would yield (42, 86, 6068)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 11615 620332 929271 900714 786900 018360 002572 218153 324114 770173 038663 175838 988464 826177 > 9⁸⁶ [i]