MinT - Best Known (32−19, 32, s)-Nets in Base 64

Best Known (32−19, 32, s)-Nets in Base 64

(32−19, 32, 184)-Net over F₆₄ — Constructive and digital

Digital (13, 32, 184)-net over F₆₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 10, 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, 22, 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]

(32−19, 32, 257)-Net over F₆₄ — Digital

Digital (13, 32, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 32, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(32−19, 32, 262)-Net in Base 64 — Constructive

(13, 32, 262)-net in base 64, using

base change [i] based on digital (5, 24, 262)-net over F₂₅₆, using
- net from sequence [i] based on digital (5, 261)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(32−19, 32, 321)-Net in Base 64

(13, 32, 321)-net in base 64, using

12 times m-reduction [i] based on (13, 44, 321)-net in base 64, using
- base change [i] based on digital (2, 33, 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]

(32−19, 32, 109567)-Net in Base 64 — Upper bound on s

There is no (13, 32, 109568)-net in base 64, because

1 times m-reduction [i] would yield (13, 31, 109568)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 98 085160 344342 341607 515893 730025 141565 760420 044464 819073 > 64³¹ [i]