MinT - Best Known (72−55, 72, s)-Nets in Base 64

Best Known (72−55, 72, s)-Nets in Base 64

(72−55, 72, 177)-Net over F₆₄ — Constructive and digital

Digital (17, 72, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 72, 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]

(72−55, 72, 216)-Net in Base 64 — Constructive

(17, 72, 216)-net in base 64, using

12 times m-reduction [i] based on (17, 84, 216)-net in base 64, using
- base change [i] based on digital (5, 72, 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]

(72−55, 72, 267)-Net over F₆₄ — Digital

Digital (17, 72, 267)-net over F₆₄, using

t-expansion [i] based on digital (16, 72, 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]

(72−55, 72, 9728)-Net in Base 64 — Upper bound on s

There is no (17, 72, 9729)-net in base 64, because

1 times m-reduction [i] would yield (17, 71, 9729)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 173 469376 669662 827279 365161 775893 781620 408808 078470 639777 385084 041086 132079 930702 452610 269354 069392 387917 449802 943373 915060 907520 > 64⁷¹ [i]