MinT - Best Known (139−36, 139, s)-Nets in Base 9

Best Known (139−36, 139, s)-Nets in Base 9

(139−36, 139, 822)-Net over F₉ — Constructive and digital

Digital (103, 139, 822)-net over F₉, using

(u,Â u+v)-construction [i] based on
1. digital (17, 35, 82)-net over F₉, using
  - base reduction for projective spaces (embedding PG(17,81) in PG(34,9)) for nets [i] based on digital (0, 18, 82)-net over F₈₁, using
    - net from sequence [i] based on digital (0, 81)-sequence over F₈₁, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 0 and N(F) ≥ 82, using
        the rational function field F₈₁(x) [i]
      - Niederreiter sequence [i]
2. digital (68, 104, 740)-net over F₉, using
  - trace code for nets [i] based on digital (16, 52, 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]

(139−36, 139, 10730)-Net over F₉ — Digital

Digital (103, 139, 10730)-net over F₉, using

Gilbertâ€“VarÅ¡amov bound [i]

(139−36, 139, large)-Net in Base 9 — Upper bound on s

There is no (103, 139, large)-net in base 9, because

34 times m-reduction [i] would yield (103, 105, large)-net in base 9, but
- mutually orthogonal hypercube bound [i]