MinT - Best Known (40, 40+55, s)-Nets in Base 27

Best Known (40, 40+55, s)-Nets in Base 27

(40, 40+55, 158)-Net over F₂₇ — Constructive and digital

Digital (40, 95, 158)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 33, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
      - a function field by SÃ©mirat [i]
2. digital (7, 62, 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]

(40, 40+55, 273)-Net over F₂₇ — Digital

Digital (40, 95, 273)-net over F₂₇, using

net from sequence [i] based on digital (40, 272)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 40 and N(F) ≥ 273, using
  - a curve from the manYPoints database [i]

(40, 40+55, 370)-Net in Base 27 — Constructive

(40, 95, 370)-net in base 27, using

1 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+55, 40415)-Net in Base 27 — Upper bound on s

There is no (40, 95, 40416)-net in base 27, because

1 times m-reduction [i] would yield (40, 94, 40416)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 353 434920 635375 682162 454139 032051 504644 848687 994718 166837 416926 658446 933911 899225 359489 070201 992878 600827 113485 115349 922379 401821 355137 > 27⁹⁴ [i]