MinT - Best Known (111−29, 111, s)-Nets in Base 9

Best Known (111−29, 111, s)-Nets in Base 9

(111−29, 111, 776)-Net over F₉ — Constructive and digital

Digital (82, 111, 776)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (7, 21, 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 (61, 90, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 45, 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]

(111−29, 111, 8580)-Net over F₉ — Digital

Digital (82, 111, 8580)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−29, 111, large)-Net in Base 9 — Upper bound on s

There is no (82, 111, large)-net in base 9, because

27 times m-reduction [i] would yield (82, 84, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]