MinT - Best Known (8, 8+19, s)-Nets in Base 64

Best Known (8, 8+19, s)-Nets in Base 64

(8, 8+19, 177)-Net over F₆₄ — Constructive and digital

Digital (8, 27, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 27, 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, 8+19, 258)-Net in Base 64 — Constructive

(8, 27, 258)-net in base 64, using

1 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]

(8, 8+19, 289)-Net in Base 64

(8, 27, 289)-net in base 64, using

1 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]

(8, 8+19, 10866)-Net in Base 64 — Upper bound on s

There is no (8, 27, 10867)-net in base 64, because

1 times m-reduction [i] would yield (8, 26, 10867)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 91364 736948 800386 503186 659502 256933 760642 974252 > 64²⁶ [i]