MinT - Best Known (57−23, 57, s)-Nets in Base 27

Best Known (57−23, 57, s)-Nets in Base 27

(57−23, 57, 218)-Net over F₂₇ — Constructive and digital

Digital (34, 57, 218)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 11, 66)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. 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]
    2. digital (1, 8, 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]
2. digital (6, 17, 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]
3. digital (6, 29, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(57−23, 57, 370)-Net in Base 27 — Constructive

(34, 57, 370)-net in base 27, using

15 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]

(57−23, 57, 1791)-Net over F₂₇ — Digital

Digital (34, 57, 1791)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−23, 57, 3655804)-Net in Base 27 — Upper bound on s

There is no (34, 57, 3655805)-net in base 27, because

1 times m-reduction [i] would yield (34, 56, 3655805)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 143 341334 573046 779498 053189 319882 566728 935432 311367 598398 544424 785442 706960 468499 > 27⁵⁶ [i]