MinT - Best Known (126−33, 126, s)-Nets in Base 9

Best Known (126−33, 126, s)-Nets in Base 9

(126−33, 126, 784)-Net over F₉ — Constructive and digital

Digital (93, 126, 784)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (12, 28, 44)-net over F₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 9, 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 (3, 19, 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 (65, 98, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 49, 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]

(126−33, 126, 788)-Net in Base 9 — Constructive

(93, 126, 788)-net in base 9, using

9¹ times duplication [i] based on (92, 125, 788)-net in base 9, using
- (u,Â u+v)-construction [i] based on
  1. (11, 27, 48)-net in base 9, using
    - base change [i] based on digital (2, 18, 48)-net over F₂₇, using
      - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
        a function field by GarcÃa and Quoos [i]
  2. digital (65, 98, 740)-net over F₉, using
    - trace code for nets [i] based on digital (16, 49, 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]

(126−33, 126, 9160)-Net over F₉ — Digital

Digital (93, 126, 9160)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−33, 126, large)-Net in Base 9 — Upper bound on s

There is no (93, 126, large)-net in base 9, because

31 times m-reduction [i] would yield (93, 95, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]