MinT - Best Known (57, 57+47, s)-Nets in Base 27

Best Known (57, 57+47, s)-Nets in Base 27

(57, 57+47, 234)-Net over F₂₇ — Constructive and digital

Digital (57, 104, 234)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 21, 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]
2. digital (6, 29, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (7, 54, 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]

(57, 57+47, 370)-Net in Base 27 — Constructive

(57, 104, 370)-net in base 27, using

t-expansion [i] based on (43, 104, 370)-net in base 27, using
- 4 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]

(57, 57+47, 1215)-Net over F₂₇ — Digital

Digital (57, 104, 1215)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+47, 932175)-Net in Base 27 — Upper bound on s

There is no (57, 104, 932176)-net in base 27, because

1 times m-reduction [i] would yield (57, 103, 932176)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2694 461743 576889 619201 154974 147767 120611 843270 774721 603784 163064 204179 765941 900037 295993 640717 196627 009633 380542 658441 573599 539279 008397 979721 641793 > 27¹⁰³ [i]