MinT - Best Known (8, 21, s)-Nets in Base 64

Best Known (8, 21, s)-Nets in Base 64

(8, 21, 177)-Net over F₆₄ — Constructive and digital

Digital (8, 21, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 21, 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]

(8, 21, 259)-Net in Base 64 — Constructive

(8, 21, 259)-net in base 64, using

3 times m-reduction [i] based on (8, 24, 259)-net in base 64, using
- base change [i] based on digital (2, 18, 259)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(8, 21, 321)-Net in Base 64

(8, 21, 321)-net in base 64, using

3 times m-reduction [i] based on (8, 24, 321)-net in base 64, using
- base change [i] based on digital (2, 18, 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]

(8, 21, 49826)-Net in Base 64 — Upper bound on s

There is no (8, 21, 49827)-net in base 64, because

1 times m-reduction [i] would yield (8, 20, 49827)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 329336 578683 237635 212942 778445 943848 > 64²⁰ [i]