MinT - Best Known (41−25, 41, s)-Nets in Base 64

Best Known (41−25, 41, s)-Nets in Base 64

(41−25, 41, 184)-Net over F₆₄ — Constructive and digital

Digital (16, 41, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 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]
2. digital (3, 28, 104)-net over F₆₄, using
  - net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(41−25, 41, 267)-Net over F₆₄ — Digital

Digital (16, 41, 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]

(41−25, 41, 288)-Net in Base 64 — Constructive

(16, 41, 288)-net in base 64, using

8 times m-reduction [i] based on (16, 49, 288)-net in base 64, using
- base change [i] based on digital (9, 42, 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]

(41−25, 41, 321)-Net in Base 64

(16, 41, 321)-net in base 64, using

15 times m-reduction [i] based on (16, 56, 321)-net in base 64, using
- base change [i] based on digital (2, 42, 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]

(41−25, 41, 88022)-Net in Base 64 — Upper bound on s

There is no (16, 41, 88023)-net in base 64, because

1 times m-reduction [i] would yield (16, 40, 88023)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 1 767016 155422 088411 832436 258764 377435 151780 127330 764904 909452 325607 241438 > 64⁴⁰ [i]