MinT - Best Known (72−56, 72, s)-Nets in Base 64

Best Known (72−56, 72, s)-Nets in Base 64

(72−56, 72, 177)-Net over F₆₄ — Constructive and digital

Digital (16, 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]

(72−56, 72, 216)-Net in Base 64 — Constructive

(16, 72, 216)-net in base 64, using

5 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]

(72−56, 72, 267)-Net over F₆₄ — Digital

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]

(72−56, 72, 7895)-Net in Base 64 — Upper bound on s

There is no (16, 72, 7896)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 11123 543792 005115 825009 301955 380277 784154 768732 898308 451936 593329 321147 590717 326867 328510 792423 779459 113626 369533 532131 956943 908536 > 64⁷² [i]