MinT - Best Known (140−79, 140, s)-Nets in Base 9

Best Known (140−79, 140, s)-Nets in Base 9

(140−79, 140, 106)-Net over F₉ — Constructive and digital

Digital (61, 140, 106)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (5, 44, 32)-net over F₉, using
  - net from sequence [i] based on digital (5, 31)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 5 and N(F) ≥ 32, using
      - a function field by GarcÃa and Quoos [i]
2. digital (17, 96, 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]

(140−79, 140, 192)-Net over F₉ — Digital

Digital (61, 140, 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]

(140−79, 140, 4821)-Net in Base 9 — Upper bound on s

There is no (61, 140, 4822)-net in base 9, because

1 times m-reduction [i] would yield (61, 139, 4822)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 4 368374 818666 525484 820087 368148 899571 680876 901488 872307 217566 845272 210338 251388 683734 250764 742087 734979 849063 237645 512969 448333 914385 > 9¹³⁹ [i]