MinT - Best Known (136−75, 136, s)-Nets in Base 9

Best Known (136−75, 136, s)-Nets in Base 9

(136−75, 136, 110)-Net over F₉ — Constructive and digital

Digital (61, 136, 110)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (7, 44, 36)-net over F₉, using
  - net from sequence [i] based on digital (7, 35)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 7 and N(F) ≥ 36, 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]

(136−75, 136, 192)-Net over F₉ — Digital

Digital (61, 136, 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]

(136−75, 136, 5530)-Net in Base 9 — Upper bound on s

There is no (61, 136, 5531)-net in base 9, because

1 times m-reduction [i] would yield (61, 135, 5531)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 668 108013 755235 918874 648041 288584 000606 939743 519673 284308 607026 277259 073786 685534 637070 184954 456895 717270 080583 909079 554585 607225 > 9¹³⁵ [i]