MinT - Best Known (75−43, 75, s)-Nets in Base 27

Best Known (75−43, 75, s)-Nets in Base 27

(75−43, 75, 146)-Net over F₂₇ — Constructive and digital

Digital (32, 75, 146)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 25, 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]
2. digital (7, 50, 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]

(75−43, 75, 223)-Net over F₂₇ — Digital

Digital (32, 75, 223)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−43, 75, 224)-Net in Base 27 — Constructive

(32, 75, 224)-net in base 27, using

1 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]

(75−43, 75, 298)-Net in Base 27

(32, 75, 298)-net in base 27, using

5 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]

(75−43, 75, 36918)-Net in Base 27 — Upper bound on s

There is no (32, 75, 36919)-net in base 27, because

1 times m-reduction [i] would yield (32, 74, 36919)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 8337 084160 300803 219487 443002 866389 566200 040934 922313 125645 070652 607383 044166 361287 579002 722636 652689 558095 > 27⁷⁴ [i]