MinT - Best Known (71−39, 71, s)-Nets in Base 27

Best Known (71−39, 71, s)-Nets in Base 27

(71−39, 71, 158)-Net over F₂₇ — Constructive and digital

Digital (32, 71, 158)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 25, 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]
2. digital (7, 46, 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]

(71−39, 71, 224)-Net in Base 27 — Constructive

(32, 71, 224)-net in base 27, using

5 times m-reduction [i] based on (32, 76, 224)-net in base 27, using
- base change [i] based on digital (13, 57, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(71−39, 71, 273)-Net over F₂₇ — Digital

Digital (32, 71, 273)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(71−39, 71, 298)-Net in Base 27

(32, 71, 298)-net in base 27, using

9 times m-reduction [i] based on (32, 80, 298)-net in base 27, using
- base change [i] based on digital (12, 60, 298)-net over F₈₁, using
  - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
      - a curve from the manYPoints database [i]

(71−39, 71, 57228)-Net in Base 27 — Upper bound on s

There is no (32, 71, 57229)-net in base 27, because

1 times m-reduction [i] would yield (32, 70, 57229)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 15685 878268 722226 088971 421725 006175 264652 541519 234398 189175 127939 939563 294402 004546 981295 749115 463731 > 27⁷⁰ [i]