MinT - Best Known (16−11, 16, s)-Nets in Base 64

Best Known (16−11, 16, s)-Nets in Base 64

(16−11, 16, 130)-Net over F₆₄ — Constructive and digital

Digital (5, 16, 130)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 5, 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, 11, 65)-net over F₆₄, using
  - net from sequence [i] based on digital (0, 64)-sequence over F₆₄ (see above)

(16−11, 16, 133)-Net over F₆₄ — Digital

Digital (5, 16, 133)-net over F₆₄, using

net from sequence [i] based on digital (5, 132)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 133, using
  - a curve from the manYPoints database [i]

(16−11, 16, 258)-Net in Base 64 — Constructive

(5, 16, 258)-net in base 64, using

base change [i] based on digital (1, 12, 258)-net over F₂₅₆, using
- net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(16−11, 16, 289)-Net in Base 64

(5, 16, 289)-net in base 64, using

base change [i] based on digital (1, 12, 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]

(16−11, 16, 10838)-Net in Base 64 — Upper bound on s

There is no (5, 16, 10839)-net in base 64, because

1 times m-reduction [i] would yield (5, 15, 10839)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1238 466030 010664 837107 747330 > 64¹⁵ [i]