MinT - Best Known (7, 22, s)-Nets in Base 64

Best Known (7, 22, s)-Nets in Base 64

(7, 22, 177)-Net over F₆₄ — Constructive and digital

Digital (7, 22, 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]

(7, 22, 258)-Net in Base 64 — Constructive

(7, 22, 258)-net in base 64, using

2 times m-reduction [i] based on (7, 24, 258)-net in base 64, using
- base change [i] based on digital (1, 18, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(7, 22, 289)-Net in Base 64

(7, 22, 289)-net in base 64, using

2 times m-reduction [i] based on (7, 24, 289)-net in base 64, using
- base change [i] based on digital (1, 18, 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]

(7, 22, 14061)-Net in Base 64 — Upper bound on s

There is no (7, 22, 14062)-net in base 64, because

1 times m-reduction [i] would yield (7, 21, 14062)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 85 105072 334238 611918 212846 566906 453820 > 64²¹ [i]