MinT - Best Known (39, 98, s)-Nets in Base 27

Best Known (39, 98, s)-Nets in Base 27

(39, 98, 140)-Net over F₂₇ — Constructive and digital

Digital (39, 98, 140)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (6, 65, 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]

(39, 98, 224)-Net in Base 27 — Constructive

(39, 98, 224)-net in base 27, using

6 times m-reduction [i] based on (39, 104, 224)-net in base 27, using
- base change [i] based on digital (13, 78, 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]

(39, 98, 271)-Net over F₂₇ — Digital

Digital (39, 98, 271)-net over F₂₇, using

net from sequence [i] based on digital (39, 270)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 39 and N(F) ≥ 271, using
  - a curve from the manYPoints database [i]

(39, 98, 298)-Net in Base 27

(39, 98, 298)-net in base 27, using

10 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]

(39, 98, 27515)-Net in Base 27 — Upper bound on s

There is no (39, 98, 27516)-net in base 27, because

1 times m-reduction [i] would yield (39, 97, 27516)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 959496 342401 612776 998682 208918 327732 661313 628403 063655 732957 499033 057430 846651 686552 399293 446685 669002 834114 485475 599214 006306 684428 157817 > 27⁹⁷ [i]