MinT - Best Known (54, 54+35, s)-Nets in Base 27

Best Known (54, 54+35, s)-Nets in Base 27

(54, 54+35, 268)-Net over F₂₇ — Constructive and digital

Digital (54, 89, 268)-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 (6, 41, 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]

(54, 54+35, 434)-Net in Base 27 — Constructive

(54, 89, 434)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (4, 21, 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. (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]

(54, 54+35, 2923)-Net over F₂₇ — Digital

Digital (54, 89, 2923)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+35, 7085716)-Net in Base 27 — Upper bound on s

There is no (54, 89, 7085717)-net in base 27, because

1 times m-reduction [i] would yield (54, 88, 7085717)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 912036 177098 609652 726978 358557 370504 473038 085822 064169 810656 412520 731276 076422 377695 903335 858342 762723 547039 309326 759157 004899 > 27⁸⁸ [i]