MinT - Best Known (52−37, 52, s)-Nets in Base 64

Best Known (52−37, 52, s)-Nets in Base 64

(52−37, 52, 177)-Net over F₆₄ — Constructive and digital

Digital (15, 52, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 52, 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]

(52−37, 52, 258)-Net over F₆₄ — Digital

Digital (15, 52, 258)-net over F₆₄, using

net from sequence [i] based on digital (15, 257)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 15 and N(F) ≥ 258, using
  - a curve from the manYPoints database [i]

(52−37, 52, 259)-Net in Base 64 — Constructive

(15, 52, 259)-net in base 64, using

base change [i] based on digital (2, 39, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(52−37, 52, 321)-Net in Base 64

(15, 52, 321)-net in base 64, using

base change [i] based on digital (2, 39, 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]

(52−37, 52, 15705)-Net in Base 64 — Upper bound on s

There is no (15, 52, 15706)-net in base 64, because

1 times m-reduction [i] would yield (15, 51, 15706)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 130 418952 207661 136078 805934 889068 976809 009820 678496 973509 515533 558810 704601 518367 461536 403052 > 64⁵¹ [i]