MinT - Best Known (51−40, 51, s)-Nets in Base 64

Best Known (51−40, 51, s)-Nets in Base 64

(51−40, 51, 177)-Net over F₆₄ — Constructive and digital

Digital (11, 51, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 51, 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]

(51−40, 51, 192)-Net in Base 64 — Constructive

(11, 51, 192)-net in base 64, using

5 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
- base change [i] based on digital (3, 48, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(51−40, 51, 225)-Net over F₆₄ — Digital

Digital (11, 51, 225)-net over F₆₄, using

t-expansion [i] based on digital (10, 51, 225)-net over F₆₄, using
- net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(51−40, 51, 5307)-Net in Base 64 — Upper bound on s

There is no (11, 51, 5308)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 130 421394 247861 756051 565105 066281 518146 171782 409475 879944 215859 910655 642229 047395 908468 235754 > 64⁵¹ [i]