MinT - Best Known (68−27, 68, s)-Nets in Base 27

Best Known (68−27, 68, s)-Nets in Base 27

(68−27, 68, 234)-Net over F₂₇ — Constructive and digital

Digital (41, 68, 234)-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 (7, 34, 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]

(68−27, 68, 370)-Net in Base 27 — Constructive

(41, 68, 370)-net in base 27, using

32 times m-reduction [i] based on (41, 100, 370)-net in base 27, using
- base change [i] based on digital (16, 75, 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]

(68−27, 68, 2261)-Net over F₂₇ — Digital

Digital (41, 68, 2261)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−27, 68, 5193509)-Net in Base 27 — Upper bound on s

There is no (41, 68, 5193510)-net in base 27, because

1 times m-reduction [i] would yield (41, 67, 5193510)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 796843 274941 519852 862565 070696 690501 454640 505390 133319 794802 238975 275463 487577 051357 397641 430053 > 27⁶⁷ [i]