MinT - Best Known (40, 40+25, s)-Nets in Base 27

Best Known (40, 40+25, s)-Nets in Base 27

(40, 40+25, 246)-Net over F₂₇ — Constructive and digital

Digital (40, 65, 246)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 14, 82)-net over F₂₇, using
  - s-reduction based on digital (6, 14, 84)-net over F₂₇, using
    - generalized (u,Â u+v)-construction [i] based on
      1. digital (0, 2, 28)-net over F₂₇, using
        kÂ =Â 2 construction [i]
      2. digital (0, 4, 28)-net over F₂₇, using
        net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
        Niederreiter sequence [i]
      3. digital (0, 8, 28)-net over F₂₇, using
        net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (7, 19, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
3. digital (7, 32, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(40, 40+25, 370)-Net in Base 27 — Constructive

(40, 65, 370)-net in base 27, using

31 times m-reduction [i] based on (40, 96, 370)-net in base 27, using
- base change [i] based on digital (16, 72, 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]

(40, 40+25, 2850)-Net over F₂₇ — Digital

Digital (40, 65, 2850)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 40+25, large)-Net in Base 27 — Upper bound on s

There is no (40, 65, large)-net in base 27, because

23 times m-reduction [i] would yield (40, 42, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]