MinT - Best Known (42, 42+29, s)-Nets in Base 27

Best Known (42, 42+29, s)-Nets in Base 27

(42, 42+29, 234)-Net over F₂₇ — Constructive and digital

Digital (42, 71, 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, 20, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
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]

(42, 42+29, 370)-Net in Base 27 — Constructive

(42, 71, 370)-net in base 27, using

33 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, 42+29, 1866)-Net over F₂₇ — Digital

Digital (42, 71, 1866)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+29, 3336585)-Net in Base 27 — Upper bound on s

There is no (42, 71, 3336586)-net in base 27, because

1 times m-reduction [i] would yield (42, 70, 3336586)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 15684 254121 947064 738571 214499 216611 739770 512358 851342 047295 741333 342885 382117 954686 375692 539006 595453 > 27⁷⁰ [i]