MinT - Best Known (96−55, 96, s)-Nets in Base 27

Best Known (96−55, 96, s)-Nets in Base 27

(96−55, 96, 164)-Net over F₂₇ — Constructive and digital

Digital (41, 96, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 34, 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, 62, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(96−55, 96, 273)-Net over F₂₇ — Digital

Digital (41, 96, 273)-net over F₂₇, using

t-expansion [i] based on digital (40, 96, 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]

(96−55, 96, 370)-Net in Base 27 — Constructive

(41, 96, 370)-net in base 27, using

4 times m-reduction [i] based on (41, 100, 370)-net in base 27, using
- base change [i] based on digital (16, 75, 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]

(96−55, 96, 45664)-Net in Base 27 — Upper bound on s

There is no (41, 96, 45665)-net in base 27, because

1 times m-reduction [i] would yield (41, 95, 45665)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 9542 344667 303648 892621 131465 908894 505418 509019 498873 076880 815476 217874 630219 378174 129524 462279 948489 021185 303512 791328 228474 588432 395843 > 27⁹⁵ [i]