MinT - Best Known (56, 56+36, s)-Nets in Base 27

Best Known (56, 56+36, s)-Nets in Base 27

(56, 56+36, 264)-Net over F₂₇ — Constructive and digital

Digital (56, 92, 264)-net over F₂₇, using

1 times m-reduction [i] based on digital (56, 93, 264)-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 (9, 27, 88)-net over F₂₇, using
    - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, 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]

(56, 56+36, 434)-Net in Base 27 — Constructive

(56, 92, 434)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (4, 22, 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. (34, 70, 370)-net in base 27, using
  - 2 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]

(56, 56+36, 3113)-Net over F₂₇ — Digital

Digital (56, 92, 3113)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+36, 6011948)-Net in Base 27 — Upper bound on s

There is no (56, 92, 6011949)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 484692 584866 016448 780879 732160 471527 928847 511656 720116 037355 916965 916508 693283 997017 021812 208377 983526 067400 007915 819897 195908 169713 > 27⁹² [i]