MinT - Best Known (28−19, 28, s)-Nets in Base 64

Best Known (28−19, 28, s)-Nets in Base 64

(28−19, 28, 177)-Net over F₆₄ — Constructive and digital

Digital (9, 28, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 28, 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]

(28−19, 28, 209)-Net over F₆₄ — Digital

Digital (9, 28, 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]

(28−19, 28, 259)-Net in Base 64 — Constructive

(9, 28, 259)-net in base 64, using

base change [i] based on digital (2, 21, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(28−19, 28, 321)-Net in Base 64

(9, 28, 321)-net in base 64, using

base change [i] based on digital (2, 21, 321)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
    - sharp bound on the number of rational points on curves with genus 2 [i]

(28−19, 28, 17252)-Net in Base 64 — Upper bound on s

There is no (9, 28, 17253)-net in base 64, because

1 times m-reduction [i] would yield (9, 27, 17253)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 5 848195 863631 214072 692680 845534 943066 732556 125424 > 64²⁷ [i]