MinT - Best Known (55, 55+47, s)-Nets in Base 27

Best Known (55, 55+47, s)-Nets in Base 27

(55, 55+47, 222)-Net over F₂₇ — Constructive and digital

Digital (55, 102, 222)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 19, 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 (6, 29, 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]
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]

(55, 55+47, 370)-Net in Base 27 — Constructive

(55, 102, 370)-net in base 27, using

t-expansion [i] based on (43, 102, 370)-net in base 27, using
- 6 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]

(55, 55+47, 1056)-Net over F₂₇ — Digital

Digital (55, 102, 1056)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+47, 699889)-Net in Base 27 — Upper bound on s

There is no (55, 102, 699890)-net in base 27, because

1 times m-reduction [i] would yield (55, 101, 699890)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 696130 714037 570659 901667 299081 342657 649918 126519 928552 262093 548421 679812 118248 780495 722565 837290 189081 914872 453576 453487 668924 050669 690269 885369 > 27¹⁰¹ [i]