MinT - Best Known (41, 41+59, s)-Nets in Base 27

Best Known (41, 41+59, s)-Nets in Base 27

(41, 41+59, 152)-Net over F₂₇ — Constructive and digital

Digital (41, 100, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 35, 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 (6, 65, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(41, 41+59, 273)-Net over F₂₇ — Digital

Digital (41, 100, 273)-net over F₂₇, using

t-expansion [i] based on digital (40, 100, 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]

(41, 41+59, 370)-Net in Base 27 — Constructive

(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]

(41, 41+59, 34541)-Net in Base 27 — Upper bound on s

There is no (41, 100, 34542)-net in base 27, because

1 times m-reduction [i] would yield (41, 99, 34542)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 5073 115387 469752 568392 449537 478568 833003 449311 007030 126628 722402 465179 255103 116451 407188 313959 424015 235156 483238 651326 781304 101562 101903 620629 > 27⁹⁹ [i]