MinT - Best Known (40, 40+41, s)-Nets in Base 9

Best Known (40, 40+41, s)-Nets in Base 9

(40, 40+41, 102)-Net over F₉ — Constructive and digital

Digital (40, 81, 102)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (3, 23, 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, 58, 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]

(40, 40+41, 159)-Net over F₉ — Digital

Digital (40, 81, 159)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+41, 6798)-Net in Base 9 — Upper bound on s

There is no (40, 81, 6799)-net in base 9, because

1 times m-reduction [i] would yield (40, 80, 6799)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 21871 804661 676501 493315 938934 168309 882939 431223 526488 730470 177509 239883 824481 > 9⁸⁰ [i]