MinT - Best Known (19, 19+27, s)-Nets in Base 27

Best Known (19, 19+27, s)-Nets in Base 27

(19, 19+27, 112)-Net over F₂₇ — Constructive and digital

Digital (19, 46, 112)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (2, 15, 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, 31, 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]

(19, 19+27, 148)-Net over F₂₇ — Digital

Digital (19, 46, 148)-net over F₂₇, using

t-expansion [i] based on digital (18, 46, 148)-net over F₂₇, using
- net from sequence [i] based on digital (18, 147)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 148, using
    - a curve from the manYPoints database [i]

(19, 19+27, 172)-Net in Base 27 — Constructive

(19, 46, 172)-net in base 27, using

2 times m-reduction [i] based on (19, 48, 172)-net in base 27, using
- base change [i] based on digital (7, 36, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(19, 19+27, 190)-Net in Base 27

(19, 46, 190)-net in base 27, using

6 times m-reduction [i] based on (19, 52, 190)-net in base 27, using
- base change [i] based on digital (6, 39, 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]

(19, 19+27, 19634)-Net in Base 27 — Upper bound on s

There is no (19, 46, 19635)-net in base 27, because

1 times m-reduction [i] would yield (19, 45, 19635)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 25801 415126 271685 145239 989136 817542 369030 215342 873200 840904 328503 > 27⁴⁵ [i]