MinT - Best Known (9, 9+23, s)-Nets in Base 64

Best Known (9, 9+23, s)-Nets in Base 64

(9, 9+23, 177)-Net over F₆₄ — Constructive and digital

Digital (9, 32, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 32, 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]

(9, 9+23, 209)-Net over F₆₄ — Digital

Digital (9, 32, 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]

(9, 9+23, 258)-Net in Base 64 — Constructive

(9, 32, 258)-net in base 64, using

base change [i] based on digital (1, 24, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(9, 9+23, 289)-Net in Base 64

(9, 32, 289)-net in base 64, using

base change [i] based on digital (1, 24, 289)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 288)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 1 and N(F) ≥ 289, using
    - sharp bound on the number of rational points on curves with genus 1 [i]

(9, 9+23, 9584)-Net in Base 64 — Upper bound on s

There is no (9, 32, 9585)-net in base 64, because

1 times m-reduction [i] would yield (9, 31, 9585)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 98 102861 787455 587562 675070 249174 140939 697213 757099 491280 > 64³¹ [i]