MinT - Best Known (72−63, 72, s)-Nets in Base 64

Best Known (72−63, 72, s)-Nets in Base 64

(72−63, 72, 177)-Net over F₆₄ — Constructive and digital

Digital (9, 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]

(72−63, 72, 209)-Net over F₆₄ — Digital

Digital (9, 72, 209)-net over F₆₄, using

net from sequence [i] based on digital (9, 208)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 9 and N(F) ≥ 209, using
  - a curve from the manYPoints database [i]

(72−63, 72, 2684)-Net in Base 64 — Upper bound on s

There is no (9, 72, 2685)-net in base 64, because

1 times m-reduction [i] would yield (9, 71, 2685)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 173 451352 329765 102307 877784 736990 309563 654049 815540 011283 172837 274273 337132 353618 420538 028646 070714 219071 971180 142168 899518 504160 > 64⁷¹ [i]