MinT - Best Known (50−29, 50, s)-Nets in Base 27

Best Known (50−29, 50, s)-Nets in Base 27

(50−29, 50, 116)-Net over F₂₇ — Constructive and digital

Digital (21, 50, 116)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 17, 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, 33, 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]

(50−29, 50, 163)-Net over F₂₇ — Digital

Digital (21, 50, 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]

(50−29, 50, 172)-Net in Base 27 — Constructive

(21, 50, 172)-net in base 27, using

6 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]

(50−29, 50, 226)-Net in Base 27

(21, 50, 226)-net in base 27, using

2 times m-reduction [i] based on (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]

(50−29, 50, 23775)-Net in Base 27 — Upper bound on s

There is no (21, 50, 23776)-net in base 27, because

1 times m-reduction [i] would yield (21, 49, 23776)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13707 590693 536390 512602 968140 106636 671767 177219 451581 238576 882503 828289 > 27⁴⁹ [i]