MinT - Best Known (42−25, 42, s)-Nets in Base 64

Best Known (42−25, 42, s)-Nets in Base 64

(42−25, 42, 193)-Net over F₆₄ — Constructive and digital

Digital (17, 42, 193)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 12, 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 (5, 30, 128)-net over F₆₄, using
  - net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
      - a function field by SÃ©mirat [i]

(42−25, 42, 267)-Net over F₆₄ — Digital

Digital (17, 42, 267)-net over F₆₄, using

t-expansion [i] based on digital (16, 42, 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]

(42−25, 42, 288)-Net in Base 64 — Constructive

(17, 42, 288)-net in base 64, using

14 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
- base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
  - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
      - a function field by SÃ©mirat [i]

(42−25, 42, 321)-Net in Base 64

(17, 42, 321)-net in base 64, using

18 times m-reduction [i] based on (17, 60, 321)-net in base 64, using
- base change [i] based on digital (2, 45, 321)-net over F₂₅₆, using
  - net from sequence [i] based on digital (2, 320)-sequence over F₂₅₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 2 and N(F) ≥ 321, using
      - sharp bound on the number of rational points on curves with genus 2 [i]

(42−25, 42, 124484)-Net in Base 64 — Upper bound on s

There is no (17, 42, 124485)-net in base 64, because

1 times m-reduction [i] would yield (17, 41, 124485)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 113 081649 713043 326265 504423 407301 210382 982959 064672 816327 243814 943678 928200 > 64⁴¹ [i]