MinT - Best Known (39, 39+55, s)-Nets in Base 27

Best Known (39, 39+55, s)-Nets in Base 27

(39, 39+55, 152)-Net over F₂₇ — Constructive and digital

Digital (39, 94, 152)-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 (6, 61, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(39, 39+55, 224)-Net in Base 27 — Constructive

(39, 94, 224)-net in base 27, using

10 times m-reduction [i] based on (39, 104, 224)-net in base 27, using
- base change [i] based on digital (13, 78, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(39, 39+55, 271)-Net over F₂₇ — Digital

Digital (39, 94, 271)-net over F₂₇, using

net from sequence [i] based on digital (39, 270)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 39 and N(F) ≥ 271, using
  - a curve from the manYPoints database [i]

(39, 39+55, 298)-Net in Base 27

(39, 94, 298)-net in base 27, using

14 times m-reduction [i] based on (39, 108, 298)-net in base 27, using
- base change [i] based on digital (12, 81, 298)-net over F₈₁, using
  - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
      - a curve from the manYPoints database [i]

(39, 39+55, 35769)-Net in Base 27 — Upper bound on s

There is no (39, 94, 35770)-net in base 27, because

1 times m-reduction [i] would yield (39, 93, 35770)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 089097 444407 818273 908119 552707 249813 031065 068345 232498 748123 616911 561595 034481 871322 643804 084822 445379 410850 733811 038467 398073 622257 > 27⁹³ [i]