MinT - Best Known (55, 55+32, s)-Nets in Base 27

Best Known (55, 55+32, s)-Nets in Base 27

(55, 55+32, 286)-Net over F₂₇ — Constructive and digital

Digital (55, 87, 286)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 12, 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 (4, 14, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (6, 22, 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]
4. digital (7, 39, 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]

(55, 55+32, 470)-Net in Base 27 — Constructive

(55, 87, 470)-net in base 27, using

1 times m-reduction [i] based on (55, 88, 470)-net in base 27, using
- base change [i] based on digital (33, 66, 470)-net over F₈₁, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 17, 100)-net over F₈₁, using
      - net from sequence [i] based on digital (1, 99)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 1 and N(F) ≥ 100, using
        a function field by SÃ©mirat [i]
    2. digital (16, 49, 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]

(55, 55+32, 4983)-Net over F₂₇ — Digital

Digital (55, 87, 4983)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+32, large)-Net in Base 27 — Upper bound on s

There is no (55, 87, large)-net in base 27, because

30 times m-reduction [i] would yield (55, 57, large)-net in base 27, but
- mutually orthogonal hypercube bound [i]