MinT - Best Known (31, 31+42, s)-Nets in Base 32

Best Known (31, 31+42, s)-Nets in Base 32

(31, 31+42, 162)-Net over F₃₂ — Constructive and digital

Digital (31, 73, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (7, 49, 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]

(31, 31+42, 273)-Net over F₃₂ — Digital

Digital (31, 73, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 73, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(31, 31+42, 288)-Net in Base 32 — Constructive

(31, 73, 288)-net in base 32, using

4 times m-reduction [i] based on (31, 77, 288)-net in base 32, using
- base change [i] based on digital (9, 55, 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]

(31, 31+42, 321)-Net in Base 32

(31, 73, 321)-net in base 32, using

4 times m-reduction [i] based on (31, 77, 321)-net in base 32, using
- base change [i] based on (9, 55, 321)-net in base 128, using
  - 1 times m-reduction [i] based on (9, 56, 321)-net in base 128, using
    - base change [i] based on digital (2, 49, 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]

(31, 31+42, 47777)-Net in Base 32 — Upper bound on s

There is no (31, 73, 47778)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 75 175581 399851 789980 952402 723028 621142 523719 613233 826454 058031 505146 289022 245432 081855 356406 395818 144913 030904 > 32⁷³ [i]