MinT - Best Known (130−69, 130, s)-Nets in Base 9

Best Known (130−69, 130, s)-Nets in Base 9

(130−69, 130, 128)-Net over F₉ — Constructive and digital

Digital (61, 130, 128)-net over F₉, using

1 times m-reduction [i] based on digital (61, 131, 128)-net over F₉, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 48, 64)-net over F₉, using
    - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
        a function field by GarcÃa and Quoos [i]
  2. digital (13, 83, 64)-net over F₉, using
    - net from sequence [i] based on digital (13, 63)-sequence over F₉ (see above)

(130−69, 130, 192)-Net over F₉ — Digital

Digital (61, 130, 192)-net over F₉, using

net from sequence [i] based on digital (61, 191)-sequence over F₉, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 61 and N(F) ≥ 192, using
  - Drinfeld modules of rank 1 [i]

(130−69, 130, 7040)-Net in Base 9 — Upper bound on s

There is no (61, 130, 7041)-net in base 9, because

1 times m-reduction [i] would yield (61, 129, 7041)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 1252 786027 497511 397753 481336 268170 624975 475023 860527 012747 829777 450012 619189 929780 182014 201059 319268 408477 716422 036844 729681 > 9¹²⁹ [i]