MinT - Best Known (54, 54+36, s)-Nets in Base 27

Best Known (54, 54+36, s)-Nets in Base 27

(54, 54+36, 258)-Net over F₂₇ — Constructive and digital

Digital (54, 90, 258)-net over F₂₇, using

1 times m-reduction [i] based on digital (54, 91, 258)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 19, 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, 25, 82)-net over F₂₇, using
    - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
  3. digital (10, 47, 94)-net over F₂₇, using
    - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
        a function field by SÃ©mirat [i]

(54, 54+36, 418)-Net in Base 27 — Constructive

(54, 90, 418)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (2, 20, 48)-net over F₂₇, using
  - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
      - a function field by GarcÃa and Quoos [i]
2. (34, 70, 370)-net in base 27, using
  - 2 times m-reduction [i] based on (34, 72, 370)-net in base 27, using
    - base change [i] based on digital (16, 54, 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+36, 2582)-Net over F₂₇ — Digital

Digital (54, 90, 2582)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+36, 4168449)-Net in Base 27 — Upper bound on s

There is no (54, 90, 4168450)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 664 873800 866747 571052 847087 210133 001951 409838 178413 692918 281405 309840 886708 678602 717538 538829 752159 042327 586367 484443 923164 745461 > 27⁹⁰ [i]