MinT - Best Known (27, 27+33, s)-Nets in Base 64

Best Known (27, 27+33, s)-Nets in Base 64

(27, 27+33, 281)-Net over F₆₄ — Constructive and digital

Digital (27, 60, 281)-net over F₆₄, using

1 times m-reduction [i] based on digital (27, 61, 281)-net over F₆₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (3, 20, 104)-net over F₆₄, using
    - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]
  2. digital (7, 41, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(27, 27+33, 327)-Net in Base 64 — Constructive

(27, 60, 327)-net in base 64, using

(u,Â u+v)-construction [i] based on
1. (4, 20, 150)-net in base 64, using
  - 1 times m-reduction [i] based on (4, 21, 150)-net in base 64, using
    - base change [i] based on digital (1, 18, 150)-net over F₁₂₈, using
      - net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]
2. digital (7, 40, 177)-net over F₆₄, using
  - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
      - a function field by GarcÃa and Quoos [i]

(27, 27+33, 511)-Net over F₆₄ — Digital

Digital (27, 60, 511)-net over F₆₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 27+33, 513)-Net in Base 64

(27, 60, 513)-net in base 64, using

16 times m-reduction [i] based on (27, 76, 513)-net in base 64, using
- base change [i] based on digital (8, 57, 513)-net over F₂₅₆, using
  - net from sequence [i] based on digital (8, 512)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 8 and N(F) ≥ 513, using
      - K_1,1 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₂₅₆ [i]

(27, 27+33, 493719)-Net in Base 64 — Upper bound on s

There is no (27, 60, 493720)-net in base 64, because

1 times m-reduction [i] would yield (27, 59, 493720)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 36696 940036 675966 995494 364987 242902 260558 148588 257287 226644 309001 848938 050873 020898 996266 080971 794535 100957 > 64⁵⁹ [i]