MinT - Best Known (39, 39+65, s)-Nets in Base 27

Best Known (39, 39+65, s)-Nets in Base 27

(39, 39+65, 116)-Net over F₂₇ — Constructive and digital

Digital (39, 104, 116)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (3, 35, 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, 69, 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]

(39, 39+65, 224)-Net in Base 27 — Constructive

(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+65, 271)-Net over F₂₇ — Digital

Digital (39, 104, 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+65, 298)-Net in Base 27

(39, 104, 298)-net in base 27, using

4 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+65, 19895)-Net in Base 27 — Upper bound on s

There is no (39, 104, 19896)-net in base 27, because

1 times m-reduction [i] would yield (39, 103, 19896)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2698 730395 875858 164977 406250 630587 982668 949405 483959 362351 427251 138468 787736 186750 911327 543585 164910 030976 478169 036228 064255 470287 708650 926257 539073 > 27¹⁰³ [i]