MinT - Best Known (21, 52, s)-Nets in Base 27

Best Known (21, 52, s)-Nets in Base 27

(21, 52, 112)-Net over F₂₇ — Constructive and digital

Digital (21, 52, 112)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (2, 17, 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. digital (4, 35, 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]

(21, 52, 163)-Net over F₂₇ — Digital

Digital (21, 52, 163)-net over F₂₇, using

net from sequence [i] based on digital (21, 162)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 21 and N(F) ≥ 163, using
  - a curve from the manYPoints database [i]

(21, 52, 172)-Net in Base 27 — Constructive

(21, 52, 172)-net in base 27, using

4 times m-reduction [i] based on (21, 56, 172)-net in base 27, using
- base change [i] based on digital (7, 42, 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]

(21, 52, 226)-Net in Base 27

(21, 52, 226)-net in base 27, using

base change [i] based on digital (8, 39, 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]

(21, 52, 18165)-Net in Base 27 — Upper bound on s

There is no (21, 52, 18166)-net in base 27, because

1 times m-reduction [i] would yield (21, 51, 18166)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 9 993245 230045 940019 992198 640664 061746 359834 300022 182410 256399 939216 351513 > 27⁵¹ [i]