MinT - Best Known (37−21, 37, s)-Nets in Base 27

Best Known (37−21, 37, s)-Nets in Base 27

(37−21, 37, 112)-Net over F₂₇ — Constructive and digital

Digital (16, 37, 112)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (2, 12, 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 (4, 25, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]

(37−21, 37, 150)-Net over F₂₇ — Digital

Digital (16, 37, 150)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(37−21, 37, 160)-Net in Base 27 — Constructive

(16, 37, 160)-net in base 27, using

7 times m-reduction [i] based on (16, 44, 160)-net in base 27, using
- base change [i] based on digital (5, 33, 160)-net over F₈₁, using
  - net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
      - a function field by SÃ©mirat [i]

(37−21, 37, 190)-Net in Base 27

(16, 37, 190)-net in base 27, using

3 times m-reduction [i] based on (16, 40, 190)-net in base 27, using
- base change [i] based on digital (6, 30, 190)-net over F₈₁, using
  - net from sequence [i] based on digital (6, 189)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 6 and N(F) ≥ 190, using
      - a curve from the manYPoints database [i]

(37−21, 37, 24764)-Net in Base 27 — Upper bound on s

There is no (16, 37, 24765)-net in base 27, because

1 times m-reduction [i] would yield (16, 36, 24765)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3382 285316 759934 323609 230590 200166 166442 117963 416113 > 27³⁶ [i]