MinT - Best Known (39, 39+63, s)-Nets in Base 27

Best Known (39, 39+63, s)-Nets in Base 27

(39, 39+63, 128)-Net over F₂₇ — Constructive and digital

Digital (39, 102, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 35, 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 (4, 67, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(39, 39+63, 224)-Net in Base 27 — Constructive

(39, 102, 224)-net in base 27, using

2 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, 39+63, 271)-Net over F₂₇ — Digital

Digital (39, 102, 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, 39+63, 298)-Net in Base 27

(39, 102, 298)-net in base 27, using

6 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, 39+63, 21989)-Net in Base 27 — Upper bound on s

There is no (39, 102, 21990)-net in base 27, because

1 times m-reduction [i] would yield (39, 101, 21990)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 698642 959295 968759 442697 704860 722828 162984 212157 150948 424558 234107 496537 076217 876874 747694 770011 436350 366679 453610 458924 127927 204282 567684 581689 > 27¹⁰¹ [i]