MinT - Best Known (40, 107, s)-Nets in Base 27

Best Known (40, 107, s)-Nets in Base 27

(40, 107, 116)-Net over F₂₇ — Constructive and digital

Digital (40, 107, 116)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 36, 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, 71, 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]

(40, 107, 224)-Net in Base 27 — Constructive

(40, 107, 224)-net in base 27, using

1 times m-reduction [i] based on (40, 108, 224)-net in base 27, using
- base change [i] based on digital (13, 81, 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]

(40, 107, 273)-Net over F₂₇ — Digital

Digital (40, 107, 273)-net over F₂₇, using

net from sequence [i] based on digital (40, 272)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 40 and N(F) ≥ 273, using
  - a curve from the manYPoints database [i]

(40, 107, 298)-Net in Base 27

(40, 107, 298)-net in base 27, using

t-expansion [i] based on (39, 107, 298)-net in base 27, using
- 1 times m-reduction [i] based on (39, 108, 298)-net in base 27, using
  - base change [i] based on digital (12, 81, 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]

(40, 107, 20031)-Net in Base 27 — Upper bound on s

There is no (40, 107, 20032)-net in base 27, because

1 times m-reduction [i] would yield (40, 106, 20032)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 049892 149325 649596 422808 051616 559742 911024 786535 751715 932751 087030 088616 866885 050264 074509 238872 483498 095101 495226 668269 736419 345509 808574 206388 691585 > 27¹⁰⁶ [i]