MinT - Best Known (40, 69, s)-Nets in Base 27

Best Known (40, 69, s)-Nets in Base 27

(40, 69, 222)-Net over F₂₇ — Constructive and digital

Digital (40, 69, 222)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 13, 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, 20, 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, 36, 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]

(40, 69, 370)-Net in Base 27 — Constructive

(40, 69, 370)-net in base 27, using

27 times m-reduction [i] based on (40, 96, 370)-net in base 27, using
- base change [i] based on digital (16, 72, 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]

(40, 69, 1477)-Net over F₂₇ — Digital

Digital (40, 69, 1477)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(40, 69, 2083632)-Net in Base 27 — Upper bound on s

There is no (40, 69, 2083633)-net in base 27, because

1 times m-reduction [i] would yield (40, 68, 2083633)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 21 514746 666483 167301 819401 827455 523137 388221 125635 985813 653099 945731 470161 057072 023029 303136 849417 > 27⁶⁸ [i]