MinT - Best Known (41−23, 41, s)-Nets in Base 32

Best Known (41−23, 41, s)-Nets in Base 32

(41−23, 41, 131)-Net over F₃₂ — Constructive and digital

Digital (18, 41, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 30, 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]

(41−23, 41, 198)-Net over F₃₂ — Digital

Digital (18, 41, 198)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(41−23, 41, 259)-Net in Base 32 — Constructive

(18, 41, 259)-net in base 32, using

1 times m-reduction [i] based on (18, 42, 259)-net in base 32, using
- base change [i] based on (11, 35, 259)-net in base 64, using
  - 1 times m-reduction [i] based on (11, 36, 259)-net in base 64, using
    - base change [i] based on digital (2, 27, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(41−23, 41, 321)-Net in Base 32

(18, 41, 321)-net in base 32, using

1 times m-reduction [i] based on (18, 42, 321)-net in base 32, using
- base change [i] based on (11, 35, 321)-net in base 64, using
  - 1 times m-reduction [i] based on (11, 36, 321)-net in base 64, using
    - base change [i] based on digital (2, 27, 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]

(41−23, 41, 47084)-Net in Base 32 — Upper bound on s

There is no (18, 41, 47085)-net in base 32, because

1 times m-reduction [i] would yield (18, 40, 47085)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 607216 906788 951240 354930 604586 866700 427918 567697 325788 319268 > 32⁴⁰ [i]