MinT - Best Known (75−57, 75, s)-Nets in Base 64

Best Known (75−57, 75, s)-Nets in Base 64

(75−57, 75, 177)-Net over F₆₄ — Constructive and digital

Digital (18, 75, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 75, 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]

(75−57, 75, 216)-Net in Base 64 — Constructive

(18, 75, 216)-net in base 64, using

16 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
- base change [i] based on digital (5, 78, 216)-net over F₁₂₈, using
  - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
      - a function field by SÃ©mirat [i]

(75−57, 75, 281)-Net over F₆₄ — Digital

Digital (18, 75, 281)-net over F₆₄, using

net from sequence [i] based on digital (18, 280)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 18 and N(F) ≥ 281, using
  - a curve from the manYPoints database [i]

(75−57, 75, 10630)-Net in Base 64 — Upper bound on s

There is no (18, 75, 10631)-net in base 64, because

1 times m-reduction [i] would yield (18, 74, 10631)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 45 436574 875797 215160 320384 150518 233565 202121 789810 822185 546409 582098 755222 124245 089827 971153 478249 085831 140343 823028 782229 651964 797296 > 64⁷⁴ [i]