MinT - Best Known (47−19, 47, s)-Nets in Base 27

Best Known (47−19, 47, s)-Nets in Base 27

(47−19, 47, 206)-Net over F₂₇ — Constructive and digital

Digital (28, 47, 206)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 2, 28)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
2. digital (0, 3, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]
3. digital (0, 3, 28)-net over F₂₇ (see above)
4. digital (0, 4, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
5. digital (0, 6, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
6. digital (0, 9, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
7. digital (1, 20, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(47−19, 47, 370)-Net in Base 27 — Constructive

(28, 47, 370)-net in base 27, using

1 times m-reduction [i] based on (28, 48, 370)-net in base 27, using
- base change [i] based on digital (16, 36, 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]

(47−19, 47, 1596)-Net over F₂₇ — Digital

Digital (28, 47, 1596)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(47−19, 47, 3300933)-Net in Base 27 — Upper bound on s

There is no (28, 47, 3300934)-net in base 27, because

1 times m-reduction [i] would yield (28, 46, 3300934)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 696199 671490 366177 111937 889241 836215 342088 876416 597430 729848 761741 > 27⁴⁶ [i]