MinT - Best Known (6, 19, s)-Nets in Base 64

Best Known (6, 19, s)-Nets in Base 64

(6, 19, 130)-Net over F₆₄ — Constructive and digital

Digital (6, 19, 130)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 6, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
      - the rational function field F₆₄(x) [i]
    - Niederreiter sequence [i]
2. digital (0, 13, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(6, 19, 161)-Net over F₆₄ — Digital

Digital (6, 19, 161)-net over F₆₄, using

net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
  - a curve from the manYPoints database [i]

(6, 19, 258)-Net in Base 64 — Constructive

(6, 19, 258)-net in base 64, using

1 times m-reduction [i] based on (6, 20, 258)-net in base 64, using
- base change [i] based on digital (1, 15, 258)-net over F₂₅₆, using
  - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
    - Niederreiter sequence [i]

(6, 19, 289)-Net in Base 64

(6, 19, 289)-net in base 64, using

1 times m-reduction [i] based on (6, 20, 289)-net in base 64, using
- base change [i] based on digital (1, 15, 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]

(6, 19, 12454)-Net in Base 64 — Upper bound on s

There is no (6, 19, 12455)-net in base 64, because

1 times m-reduction [i] would yield (6, 18, 12455)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 324 575748 971241 832743 816174 572809 > 64¹⁸ [i]