MinT - Best Known (125−37, 125, s)-Nets in Base 9

Best Known (125−37, 125, s)-Nets in Base 9

(125−37, 125, 756)-Net over F₉ — Constructive and digital

Digital (88, 125, 756)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (1, 19, 16)-net over F₉, using
  - net from sequence [i] based on digital (1, 15)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 1 and N(F) ≥ 16, using
      - a function field by GarcÃa and Quoos [i]
2. digital (69, 106, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 53, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(125−37, 125, 3691)-Net over F₉ — Digital

Digital (88, 125, 3691)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−37, 125, 3537605)-Net in Base 9 — Upper bound on s

There is no (88, 125, 3537606)-net in base 9, because

1 times m-reduction [i] would yield (88, 124, 3537606)-net in base 9, but
- the generalized Rao bound for nets shows that 9^mÂ â‰¥ 21187 163276 441421 572539 388148 254611 657505 016128 414697 633613 805854 661206 293798 984037 779828 333892 973069 968113 038560 815777 > 9¹²⁴ [i]