MinT - Best Known (32−21, 32, s)-Nets in Base 64

Best Known (32−21, 32, s)-Nets in Base 64

(32−21, 32, 177)-Net over F₆₄ — Constructive and digital

Digital (11, 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]

(32−21, 32, 225)-Net over F₆₄ — Digital

Digital (11, 32, 225)-net over F₆₄, using

t-expansion [i] based on digital (10, 32, 225)-net over F₆₄, using
- net from sequence [i] based on digital (10, 224)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 10 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(32−21, 32, 260)-Net in Base 64 — Constructive

(11, 32, 260)-net in base 64, using

base change [i] based on digital (3, 24, 260)-net over F₂₅₆, using
- net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(32−21, 32, 321)-Net in Base 64

(11, 32, 321)-net in base 64, using

4 times m-reduction [i] based on (11, 36, 321)-net in base 64, using
- base change [i] based on digital (2, 27, 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]

(32−21, 32, 28557)-Net in Base 64 — Upper bound on s

There is no (11, 32, 28558)-net in base 64, because

1 times m-reduction [i] would yield (11, 31, 28558)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 98 090905 793003 075936 510747 414413 511636 605248 713517 926590 > 64³¹ [i]