MinT - Best Known (40, 40+53, s)-Nets in Base 27

Best Known (40, 40+53, s)-Nets in Base 27

(40, 40+53, 164)-Net over F₂₇ — Constructive and digital

Digital (40, 93, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 33, 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]
2. digital (7, 60, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(40, 40+53, 273)-Net over F₂₇ — Digital

Digital (40, 93, 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+53, 370)-Net in Base 27 — Constructive

(40, 93, 370)-net in base 27, using

3 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+53, 47099)-Net in Base 27 — Upper bound on s

There is no (40, 93, 47100)-net in base 27, because

1 times m-reduction [i] would yield (40, 92, 47100)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 484693 722516 053745 536124 230409 655918 160818 386115 538587 000262 162721 571290 055210 227755 434133 579702 711136 515973 262086 542193 215311 320585 > 27⁹² [i]