MinT - Best Known (34, 34+51, s)-Nets in Base 27

Best Known (34, 34+51, s)-Nets in Base 27

(34, 34+51, 132)-Net over F₂₇ — Constructive and digital

Digital (34, 85, 132)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 29, 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 (5, 56, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]

(34, 34+51, 172)-Net in Base 27 — Constructive

(34, 85, 172)-net in base 27, using

23 times m-reduction [i] based on (34, 108, 172)-net in base 27, using
- base change [i] based on digital (7, 81, 172)-net over F₈₁, using
  - net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
      - a function field by SÃ©mirat [i]

(34, 34+51, 220)-Net over F₂₇ — Digital

Digital (34, 85, 220)-net over F₂₇, using

t-expansion [i] based on digital (33, 85, 220)-net over F₂₇, using
- net from sequence [i] based on digital (33, 219)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 33 and N(F) ≥ 220, using
    - Drinfeld modules of rank 1 [i]

(34, 34+51, 298)-Net in Base 27

(34, 85, 298)-net in base 27, using

3 times m-reduction [i] based on (34, 88, 298)-net in base 27, using
- base change [i] based on digital (12, 66, 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]

(34, 34+51, 25223)-Net in Base 27 — Upper bound on s

There is no (34, 85, 25224)-net in base 27, because

1 times m-reduction [i] would yield (34, 84, 25224)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 716382 267861 251583 681678 682472 385880 198575 676251 386659 259527 784096 748256 633839 717390 365882 030400 340212 273970 590908 501585 > 27⁸⁴ [i]