MinT - Best Known (67−27, 67, s)-Nets in Base 27

Best Known (67−27, 67, s)-Nets in Base 27

(67−27, 67, 228)-Net over F₂₇ — Constructive and digital

Digital (40, 67, 228)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 15, 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, 19, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (6, 33, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(67−27, 67, 370)-Net in Base 27 — Constructive

(40, 67, 370)-net in base 27, using

29 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]

(67−27, 67, 1994)-Net over F₂₇ — Digital

Digital (40, 67, 1994)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−27, 67, 4030471)-Net in Base 27 — Upper bound on s

There is no (40, 67, 4030472)-net in base 27, because

1 times m-reduction [i] would yield (40, 66, 4030472)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 29512 712137 948064 165786 848761 856794 423689 731093 853865 603827 062891 075881 536538 146457 297843 314385 > 27⁶⁶ [i]