MinT - Best Known (34, 34+39, s)-Nets in Base 27

Best Known (34, 34+39, s)-Nets in Base 27

(34, 34+39, 166)-Net over F₂₇ — Constructive and digital

Digital (34, 73, 166)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 26, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (8, 47, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
      - a function field by SÃ©mirat [i]

(34, 34+39, 224)-Net in Base 27 — Constructive

(34, 73, 224)-net in base 27, using

11 times m-reduction [i] based on (34, 84, 224)-net in base 27, using
- base change [i] based on digital (13, 63, 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]

(34, 34+39, 328)-Net over F₂₇ — Digital

Digital (34, 73, 328)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(34, 34+39, 80966)-Net in Base 27 — Upper bound on s

There is no (34, 73, 80967)-net in base 27, because

1 times m-reduction [i] would yield (34, 72, 80967)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 11 435193 250255 026870 941236 847489 180552 705548 081820 036079 374233 974025 920426 727997 213890 716399 702790 404979 > 27⁷² [i]