MinT - Best Known (54, 54+47, s)-Nets in Base 27

Best Known (54, 54+47, s)-Nets in Base 27

(54, 54+47, 216)-Net over F₂₇ — Constructive and digital

Digital (54, 101, 216)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 19, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (6, 29, 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]
3. digital (6, 53, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(54, 54+47, 370)-Net in Base 27 — Constructive

(54, 101, 370)-net in base 27, using

t-expansion [i] based on (43, 101, 370)-net in base 27, using
- 7 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 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]

(54, 54+47, 985)-Net over F₂₇ — Digital

Digital (54, 101, 985)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+47, 606450)-Net in Base 27 — Upper bound on s

There is no (54, 101, 606451)-net in base 27, because

1 times m-reduction [i] would yield (54, 100, 606451)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 136895 698425 830871 657359 787622 818743 304008 463151 429850 354435 966822 562243 030023 359574 112327 157361 084715 891311 654550 376563 094525 596014 527357 351323 > 27¹⁰⁰ [i]