MinT - Best Known (55, 55+37, s)-Nets in Base 27

Best Known (55, 55+37, s)-Nets in Base 27

(55, 55+37, 260)-Net over F₂₇ — Constructive and digital

Digital (55, 92, 260)-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 (8, 26, 84)-net over F₂₇, using
  - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, 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, 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]

(55, 55+37, 422)-Net in Base 27 — Constructive

(55, 92, 422)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, 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]

(55, 55+37, 2517)-Net over F₂₇ — Digital

Digital (55, 92, 2517)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(55, 55+37, 5006047)-Net in Base 27 — Upper bound on s

There is no (55, 92, 5006048)-net in base 27, because

1 times m-reduction [i] would yield (55, 91, 5006048)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 17951 625034 301173 333487 820677 876666 221008 652290 689729 928106 262785 947454 399588 332033 966983 459191 959095 895871 500953 874055 936912 469313 > 27⁹¹ [i]