MinT - Best Known (59−43, 59, s)-Nets in Base 64

Best Known (59−43, 59, s)-Nets in Base 64

(59−43, 59, 177)-Net over F₆₄ — Constructive and digital

Digital (16, 59, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 59, 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]

(59−43, 59, 258)-Net in Base 64 — Constructive

(16, 59, 258)-net in base 64, using

1 times m-reduction [i] based on (16, 60, 258)-net in base 64, using
- base change [i] based on digital (1, 45, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(59−43, 59, 267)-Net over F₆₄ — Digital

Digital (16, 59, 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]

(59−43, 59, 289)-Net in Base 64

(16, 59, 289)-net in base 64, using

1 times m-reduction [i] based on (16, 60, 289)-net in base 64, using
- base change [i] based on digital (1, 45, 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]

(59−43, 59, 13406)-Net in Base 64 — Upper bound on s

There is no (16, 59, 13407)-net in base 64, because

1 times m-reduction [i] would yield (16, 58, 13407)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 573 848065 499828 417248 812523 500315 985708 298525 373030 017897 417008 923727 525073 063412 687106 659899 172678 530698 > 64⁵⁸ [i]