MinT - Best Known (89−71, 89, s)-Nets in Base 64

Best Known (89−71, 89, s)-Nets in Base 64

(89−71, 89, 177)-Net over F₆₄ — Constructive and digital

Digital (18, 89, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 89, 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]

(89−71, 89, 216)-Net in Base 64 — Constructive

(18, 89, 216)-net in base 64, using

2 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]

(89−71, 89, 281)-Net over F₆₄ — Digital

Digital (18, 89, 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]

(89−71, 89, 7659)-Net in Base 64 — Upper bound on s

There is no (18, 89, 7660)-net in base 64, because

1 times m-reduction [i] would yield (18, 88, 7660)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 882 012684 374178 773804 353424 302192 055784 471900 342958 013748 104021 404769 632289 145098 780468 364973 662713 149106 832347 420411 102333 865891 189234 988844 191734 591445 867405 > 64⁸⁸ [i]