MinT - Best Known (41, 41+47, s)-Nets in Base 27

Best Known (41, 41+47, s)-Nets in Base 27

(41, 41+47, 178)-Net over F₂₇ — Constructive and digital

Digital (41, 88, 178)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (8, 31, 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]
2. digital (10, 57, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(41, 41+47, 370)-Net in Base 27 — Constructive

(41, 88, 370)-net in base 27, using

12 times m-reduction [i] based on (41, 100, 370)-net in base 27, using
- base change [i] based on digital (16, 75, 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]

(41, 41+47, 378)-Net over F₂₇ — Digital

Digital (41, 88, 378)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(41, 41+47, 94127)-Net in Base 27 — Upper bound on s

There is no (41, 88, 94128)-net in base 27, because

1 times m-reduction [i] would yield (41, 87, 94128)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 33782 108859 827299 452760 764417 941675 060818 679585 902641 154426 048505 268899 935019 815042 530789 715527 875795 616982 888115 304819 399873 > 27⁸⁷ [i]