MinT - Best Known (15, 15+23, s)-Nets in Base 64

Best Known (15, 15+23, s)-Nets in Base 64

(15, 15+23, 184)-Net over F₆₄ — Constructive and digital

Digital (15, 38, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. 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]
2. digital (3, 26, 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]

(15, 15+23, 258)-Net over F₆₄ — Digital

Digital (15, 38, 258)-net over F₆₄, using

net from sequence [i] based on digital (15, 257)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 15 and N(F) ≥ 258, using
  - a curve from the manYPoints database [i]

(15, 15+23, 288)-Net in Base 64 — Constructive

(15, 38, 288)-net in base 64, using

4 times m-reduction [i] based on (15, 42, 288)-net in base 64, using
- base change [i] based on digital (9, 36, 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]

(15, 15+23, 321)-Net in Base 64

(15, 38, 321)-net in base 64, using

14 times m-reduction [i] based on (15, 52, 321)-net in base 64, using
- base change [i] based on digital (2, 39, 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]

(15, 15+23, 92679)-Net in Base 64 — Upper bound on s

There is no (15, 38, 92680)-net in base 64, because

1 times m-reduction [i] would yield (15, 37, 92680)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 6 740673 927978 686146 059960 601655 413306 258654 850431 936923 247051 379776 > 64³⁷ [i]