MinT - Best Known (53, 53+35, s)-Nets in Base 27

Best Known (53, 53+35, s)-Nets in Base 27

(53, 53+35, 260)-Net over F₂₇ — Constructive and digital

Digital (53, 88, 260)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 12, 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]
2. digital (4, 15, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (4, 21, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
4. digital (5, 40, 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]

(53, 53+35, 422)-Net in Base 27 — Constructive

(53, 88, 422)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (3, 20, 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. (33, 68, 370)-net in base 27, using
  - base change [i] based on digital (16, 51, 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]

(53, 53+35, 2655)-Net over F₂₇ — Digital

Digital (53, 88, 2655)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+35, 5836947)-Net in Base 27 — Upper bound on s

There is no (53, 88, 5836948)-net in base 27, because

1 times m-reduction [i] would yield (53, 87, 5836948)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 33779 122188 808305 484681 626719 660301 544427 118193 123380 797537 033366 217272 142483 513461 895008 926019 099886 458902 104592 769831 553033 > 27⁸⁷ [i]