MinT - Best Known (90−40, 90, s)-Nets in Base 27

Best Known (90−40, 90, s)-Nets in Base 27

(90−40, 90, 222)-Net over F₂₇ — Constructive and digital

Digital (50, 90, 222)-net over F₂₇, using

1 times m-reduction [i] based on digital (50, 91, 222)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 17, 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, 26, 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, 48, 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]

(90−40, 90, 370)-Net in Base 27 — Constructive

(50, 90, 370)-net in base 27, using

t-expansion [i] based on (43, 90, 370)-net in base 27, using
- 18 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]

(90−40, 90, 1210)-Net over F₂₇ — Digital

Digital (50, 90, 1210)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−40, 90, 881992)-Net in Base 27 — Upper bound on s

There is no (50, 90, 881993)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 664 884548 072196 157302 730287 095122 763039 817397 833148 733554 801995 896230 542773 103632 662875 344550 222989 714021 286118 979463 685090 127217 > 27⁹⁰ [i]