MinT - Best Known (44−26, 44, s)-Nets in Base 64

Best Known (44−26, 44, s)-Nets in Base 64

(44−26, 44, 193)-Net over F₆₄ — Constructive and digital

Digital (18, 44, 193)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (0, 13, 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, 31, 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]

(44−26, 44, 281)-Net over F₆₄ — Digital

Digital (18, 44, 281)-net over F₆₄, using

net from sequence [i] based on digital (18, 280)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 18 and N(F) ≥ 281, using
  - a curve from the manYPoints database [i]

(44−26, 44, 288)-Net in Base 64 — Constructive

(18, 44, 288)-net in base 64, using

19 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
- base change [i] based on digital (9, 54, 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]

(44−26, 44, 321)-Net in Base 64

(18, 44, 321)-net in base 64, using

20 times m-reduction [i] based on (18, 64, 321)-net in base 64, using
- base change [i] based on digital (2, 48, 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]

(44−26, 44, 116752)-Net in Base 64 — Upper bound on s

There is no (18, 44, 116753)-net in base 64, because

the generalized Rao bound for nets shows that 64^mÂ â‰¥ 29 643642 616144 908054 624153 918605 143562 426593 120099 712709 435266 228201 838169 746640 > 64⁴⁴ [i]