MinT - Best Known (6, 6+11, s)-Nets in Base 64

Best Known (6, 6+11, s)-Nets in Base 64

(6, 6+11, 145)-Net over F₆₄ — Constructive and digital

Digital (6, 17, 145)-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 (1, 12, 80)-net over F₆₄, using
  - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
      - a function field by SÃ©mirat [i]

(6, 6+11, 161)-Net over F₆₄ — Digital

Digital (6, 17, 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, 6+11, 258)-Net in Base 64 — Constructive

(6, 17, 258)-net in base 64, using

3 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, 6+11, 289)-Net in Base 64

(6, 17, 289)-net in base 64, using

3 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, 6+11, 24902)-Net in Base 64 — Upper bound on s

There is no (6, 17, 24903)-net in base 64, because

1 times m-reduction [i] would yield (6, 16, 24903)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 79243 043525 807470 791571 972954 > 64¹⁶ [i]