MinT - Best Known (72−30, 72, s)-Nets in Base 27

Best Known (72−30, 72, s)-Nets in Base 27

(72−30, 72, 222)-Net over F₂₇ — Constructive and digital

Digital (42, 72, 222)-net over F₂₇, using

1 times m-reduction [i] based on digital (42, 73, 222)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 14, 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, 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]
  3. digital (7, 38, 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]

(72−30, 72, 370)-Net in Base 27 — Constructive

(42, 72, 370)-net in base 27, using

32 times m-reduction [i] based on (42, 104, 370)-net in base 27, using
- base change [i] based on digital (16, 78, 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]

(72−30, 72, 1621)-Net over F₂₇ — Digital

Digital (42, 72, 1621)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−30, 72, 1833739)-Net in Base 27 — Upper bound on s

There is no (42, 72, 1833740)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 11 433887 193778 841272 282786 302784 520274 961450 759466 041026 859678 252434 649850 613417 507176 981166 723051 645873 > 27⁷² [i]