MinT - Best Known (56−33, 56, s)-Nets in Base 32

Best Known (56−33, 56, s)-Nets in Base 32

(56−33, 56, 131)-Net over F₃₂ — Constructive and digital

Digital (23, 56, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 16, 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, 40, 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]

(56−33, 56, 185)-Net over F₃₂ — Digital

Digital (23, 56, 185)-net over F₃₂, using

t-expansion [i] based on digital (21, 56, 185)-net over F₃₂, using
- net from sequence [i] based on digital (21, 184)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 21 and N(F) ≥ 185, using
    - a curve from the manYPoints database [i]

(56−33, 56, 259)-Net in Base 32 — Constructive

(23, 56, 259)-net in base 32, using

base change [i] based on digital (2, 35, 259)-net over F₂₅₆, using
- net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
  - Niederreiter sequence [i]

(56−33, 56, 321)-Net in Base 32

(23, 56, 321)-net in base 32, using

base change [i] based on digital (2, 35, 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]

(56−33, 56, 32735)-Net in Base 32 — Upper bound on s

There is no (23, 56, 32736)-net in base 32, because

1 times m-reduction [i] would yield (23, 55, 32736)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 60712 559458 496663 061778 100490 860178 298860 909852 426966 644417 929890 637141 237584 643311 > 32⁵⁵ [i]