MinT - Best Known (50, 50+44, s)-Nets in Base 27

Best Known (50, 50+44, s)-Nets in Base 27

(50, 50+44, 204)-Net over F₂₇ — Constructive and digital

Digital (50, 94, 204)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 18, 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, 26, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (6, 50, 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]

(50, 50+44, 370)-Net in Base 27 — Constructive

(50, 94, 370)-net in base 27, using

t-expansion [i] based on (43, 94, 370)-net in base 27, using
- 14 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(50, 50+44, 896)-Net over F₂₇ — Digital

Digital (50, 94, 896)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(50, 50+44, 454669)-Net in Base 27 — Upper bound on s

There is no (50, 94, 454670)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 353 349172 189258 003626 627469 526294 932812 480144 420264 400599 645517 053068 503151 746044 542749 125732 909770 231711 584632 764150 365230 424014 998757 > 27⁹⁴ [i]