MinT - Best Known (59−34, 59, s)-Nets in Base 32

Best Known (59−34, 59, s)-Nets in Base 32

(59−34, 59, 142)-Net over F₃₂ — Constructive and digital

Digital (25, 59, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 18, 44)-net over F₃₂, using
  - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
      - a function field by SÃ©mirat [i]
2. digital (7, 41, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]

(59−34, 59, 225)-Net over F₃₂ — Digital

Digital (25, 59, 225)-net over F₃₂, using

t-expansion [i] based on digital (24, 59, 225)-net over F₃₂, using
- net from sequence [i] based on digital (24, 224)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 24 and N(F) ≥ 225, using
    - a curve from the manYPoints database [i]

(59−34, 59, 259)-Net in Base 32 — Constructive

(25, 59, 259)-net in base 32, using

1 times m-reduction [i] based on (25, 60, 259)-net in base 32, using
- base change [i] based on (15, 50, 259)-net in base 64, using
  - 2 times m-reduction [i] based on (15, 52, 259)-net in base 64, using
    - base change [i] based on digital (2, 39, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(59−34, 59, 321)-Net in Base 32

(25, 59, 321)-net in base 32, using

1 times m-reduction [i] based on (25, 60, 321)-net in base 32, using
- base change [i] based on (15, 50, 321)-net in base 64, using
  - 2 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]

(59−34, 59, 38747)-Net in Base 32 — Upper bound on s

There is no (25, 59, 38748)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 63671 747377 251981 181957 644916 737193 275103 093583 749623 344381 875712 735439 704122 239148 695143 > 32⁵⁹ [i]