MinT - Best Known (53−21, 53, s)-Nets in Base 27

Best Known (53−21, 53, s)-Nets in Base 27

(53−21, 53, 210)-Net over F₂₇ — Constructive and digital

Digital (32, 53, 210)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 11, 66)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 3, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
        Niederreiter sequence [i]
    2. digital (1, 8, 38)-net over F₂₇, using
      - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]
2. digital (5, 15, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]
3. digital (6, 27, 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]

(53−21, 53, 370)-Net in Base 27 — Constructive

(32, 53, 370)-net in base 27, using

11 times m-reduction [i] based on (32, 64, 370)-net in base 27, using
- base change [i] based on digital (16, 48, 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]

(53−21, 53, 1994)-Net over F₂₇ — Digital

Digital (32, 53, 1994)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(53−21, 53, 4831634)-Net in Base 27 — Upper bound on s

There is no (32, 53, 4831635)-net in base 27, because

1 times m-reduction [i] would yield (32, 52, 4831635)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 269 721783 930969 229024 391441 183818 743950 699301 946881 479666 274167 351373 942533 > 27⁵² [i]