MinT - Best Known (42, 69, s)-Nets in Base 27

Best Known (42, 69, s)-Nets in Base 27

(42, 69, 240)-Net over F₂₇ — Constructive and digital

Digital (42, 69, 240)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (2, 8, 48)-net over F₂₇, using
  - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
      - a function field by GarcÃa and Quoos [i]
2. 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]
3. digital (4, 17, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
4. digital (4, 31, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(42, 69, 370)-Net in Base 27 — Constructive

(42, 69, 370)-net in base 27, using

35 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]

(42, 69, 2565)-Net over F₂₇ — Digital

Digital (42, 69, 2565)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 69, 6692154)-Net in Base 27 — Upper bound on s

There is no (42, 69, 6692155)-net in base 27, because

1 times m-reduction [i] would yield (42, 68, 6692155)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 21 514763 564760 253991 676141 479928 718280 859478 712095 246292 556002 628726 336990 707317 975523 769831 837639 > 27⁶⁸ [i]