MinT - Best Known (20, 20+27, s)-Nets in Base 27

Best Known (20, 20+27, s)-Nets in Base 27

(20, 20+27, 116)-Net over F₂₇ — Constructive and digital

Digital (20, 47, 116)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by SÃ©mirat [i]
2. digital (4, 31, 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]

(20, 20+27, 159)-Net over F₂₇ — Digital

Digital (20, 47, 159)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(20, 20+27, 172)-Net in Base 27 — Constructive

(20, 47, 172)-net in base 27, using

5 times m-reduction [i] based on (20, 52, 172)-net in base 27, using
- base change [i] based on digital (7, 39, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(20, 20+27, 226)-Net in Base 27

(20, 47, 226)-net in base 27, using

1 times m-reduction [i] based on (20, 48, 226)-net in base 27, using
- base change [i] based on digital (8, 36, 226)-net over F₈₁, using
  - net from sequence [i] based on digital (8, 225)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 8 and N(F) ≥ 226, using
      - a curve from the manYPoints database [i]

(20, 20+27, 25301)-Net in Base 27 — Upper bound on s

There is no (20, 47, 25302)-net in base 27, because

1 times m-reduction [i] would yield (20, 46, 25302)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 696364 133579 168945 917141 041849 308083 491710 969600 101538 396082 213765 > 27⁴⁶ [i]