MinT - Best Known (47, 47+31, s)-Nets in Base 27

Best Known (47, 47+31, s)-Nets in Base 27

(47, 47+31, 248)-Net over F₂₇ — Constructive and digital

Digital (47, 78, 248)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 10, 56)-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 (0, 7, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (4, 14, 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, 19, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
4. digital (4, 35, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(47, 47+31, 398)-Net in Base 27 — Constructive

(47, 78, 398)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (0, 15, 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. (32, 63, 370)-net in base 27, using
  - 1 times m-reduction [i] based on (32, 64, 370)-net in base 27, using
    - base change [i] based on digital (16, 48, 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, 47+31, 2455)-Net over F₂₇ — Digital

Digital (47, 78, 2455)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(47, 47+31, 5501232)-Net in Base 27 — Upper bound on s

There is no (47, 78, 5501233)-net in base 27, because

1 times m-reduction [i] would yield (47, 77, 5501233)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 164 062820 489852 406982 515121 806127 167473 265912 554298 347381 655235 413879 644109 184141 693275 205997 546256 553705 774995 > 27⁷⁷ [i]