MinT - Best Known (57, 94, s)-Nets in Base 27

Best Known (57, 94, s)-Nets in Base 27

(57, 94, 270)-Net over F₂₇ — Constructive and digital

Digital (57, 94, 270)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 19, 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]
2. digital (10, 28, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]
3. digital (10, 47, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(57, 94, 438)-Net in Base 27 — Constructive

(57, 94, 438)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (5, 23, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]
2. (34, 71, 370)-net in base 27, using
  - 1 times m-reduction [i] based on (34, 72, 370)-net in base 27, using
    - base change [i] based on digital (16, 54, 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]

(57, 94, 3019)-Net over F₂₇ — Digital

Digital (57, 94, 3019)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 94, 7219973)-Net in Base 27 — Upper bound on s

There is no (57, 94, 7219974)-net in base 27, because

1 times m-reduction [i] would yield (57, 93, 7219974)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 13 086713 696462 342935 271494 318441 321703 666856 587079 821388 634399 803249 199455 439208 429608 671631 936693 260177 756937 865263 821814 790594 298573 > 27⁹³ [i]