MinT - Best Known (62−45, 62, s)-Nets in Base 64

Best Known (62−45, 62, s)-Nets in Base 64

(62−45, 62, 177)-Net over F₆₄ — Constructive and digital

Digital (17, 62, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 62, 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]

(62−45, 62, 258)-Net in Base 64 — Constructive

(17, 62, 258)-net in base 64, using

2 times m-reduction [i] based on (17, 64, 258)-net in base 64, using
- base change [i] based on digital (1, 48, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(62−45, 62, 267)-Net over F₆₄ — Digital

Digital (17, 62, 267)-net over F₆₄, using

t-expansion [i] based on digital (16, 62, 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]

(62−45, 62, 289)-Net in Base 64

(17, 62, 289)-net in base 64, using

2 times m-reduction [i] based on (17, 64, 289)-net in base 64, using
- base change [i] based on digital (1, 48, 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]

(62−45, 62, 14629)-Net in Base 64 — Upper bound on s

There is no (17, 62, 14630)-net in base 64, because

1 times m-reduction [i] would yield (17, 61, 14630)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 150 377313 024533 598359 544738 739592 982777 975070 728786 064591 027357 040639 162772 231923 146197 973640 153872 390146 302416 > 64⁶¹ [i]