MinT - Best Known (89, 89+31, s)-Nets in Base 9

Best Known (89, 89+31, s)-Nets in Base 9

(89, 89+31, 784)-Net over F₉ — Constructive and digital

Digital (89, 120, 784)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (11, 26, 44)-net over F₉, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 8, 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, 18, 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 (63, 94, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 47, 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]

(89, 89+31, 788)-Net in Base 9 — Constructive

(89, 120, 788)-net in base 9, using

(u,Â u+v)-construction [i] based on
1. (11, 26, 48)-net in base 9, using
  - 1 times m-reduction [i] based on (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 (63, 94, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 47, 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]

(89, 89+31, 9893)-Net over F₉ — Digital

Digital (89, 120, 9893)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+31, large)-Net in Base 9 — Upper bound on s

There is no (89, 120, large)-net in base 9, because

29 times m-reduction [i] would yield (89, 91, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]