MinT - Best Known (55−39, 55, s)-Nets in Base 64

Best Known (55−39, 55, s)-Nets in Base 64

(55−39, 55, 177)-Net over F₆₄ — Constructive and digital

Digital (16, 55, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 55, 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]

(55−39, 55, 259)-Net in Base 64 — Constructive

(16, 55, 259)-net in base 64, using

1 times m-reduction [i] based on (16, 56, 259)-net in base 64, using
- base change [i] based on digital (2, 42, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(55−39, 55, 267)-Net over F₆₄ — Digital

Digital (16, 55, 267)-net over F₆₄, using

net from sequence [i] based on digital (16, 266)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 16 and N(F) ≥ 267, using
  - a curve from the manYPoints database [i]

(55−39, 55, 321)-Net in Base 64

(16, 55, 321)-net in base 64, using

1 times m-reduction [i] based on (16, 56, 321)-net in base 64, using
- base change [i] based on digital (2, 42, 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]

(55−39, 55, 17099)-Net in Base 64 — Upper bound on s

There is no (16, 55, 17100)-net in base 64, because

1 times m-reduction [i] would yield (16, 54, 17100)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 34 181307 446168 389303 860885 469408 254415 188285 966611 044663 979669 555286 098381 094819 180165 409078 437556 > 64⁵⁴ [i]