MinT - Best Known (25−17, 25, s)-Nets in Base 64

Best Known (25−17, 25, s)-Nets in Base 64

(25−17, 25, 177)-Net over F₆₄ — Constructive and digital

Digital (8, 25, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 25, 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]

(25−17, 25, 258)-Net in Base 64 — Constructive

(8, 25, 258)-net in base 64, using

3 times m-reduction [i] based on (8, 28, 258)-net in base 64, using
- base change [i] based on digital (1, 21, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(25−17, 25, 289)-Net in Base 64

(8, 25, 289)-net in base 64, using

3 times m-reduction [i] based on (8, 28, 289)-net in base 64, using
- base change [i] based on digital (1, 21, 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]

(25−17, 25, 15659)-Net in Base 64 — Upper bound on s

There is no (8, 25, 15660)-net in base 64, because

1 times m-reduction [i] would yield (8, 24, 15660)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 22 301931 957999 865635 259745 785210 861813 503457 > 64²⁴ [i]