MinT - Best Known (65−49, 65, s)-Nets in Base 64

Best Known (65−49, 65, s)-Nets in Base 64

(65−49, 65, 177)-Net over F₆₄ — Constructive and digital

Digital (16, 65, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 65, 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]

(65−49, 65, 216)-Net in Base 64 — Constructive

(16, 65, 216)-net in base 64, using

12 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
- base change [i] based on digital (5, 66, 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]

(65−49, 65, 267)-Net over F₆₄ — Digital

Digital (16, 65, 267)-net over F₆₄, using

net from sequence [i] based on digital (16, 266)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 16 and N(F) ≥ 267, using
  - a curve from the manYPoints database [i]

(65−49, 65, 10185)-Net in Base 64 — Upper bound on s

There is no (16, 65, 10186)-net in base 64, because

1 times m-reduction [i] would yield (16, 64, 10186)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 39 418779 853939 213772 621381 718927 418660 655675 820468 922201 437311 382444 130198 938963 284632 108134 076339 321096 250971 187356 > 64⁶⁴ [i]