MinT - Best Known (56−31, 56, s)-Nets in Base 32

Best Known (56−31, 56, s)-Nets in Base 32

(56−31, 56, 162)-Net over F₃₂ — Constructive and digital

Digital (25, 56, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 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, 38, 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]

(56−31, 56, 258)-Net over F₃₂ — Digital

Digital (25, 56, 258)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−31, 56, 288)-Net in Base 32 — Constructive

(25, 56, 288)-net in base 32, using

base change [i] based on digital (9, 40, 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]

(56−31, 56, 321)-Net in Base 32

(25, 56, 321)-net in base 32, using

4 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]

(56−31, 56, 68429)-Net in Base 32 — Upper bound on s

There is no (25, 56, 68430)-net in base 32, because

1 times m-reduction [i] would yield (25, 55, 68430)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 60720 633100 814447 110968 403286 422340 478145 150233 061149 004108 991814 041975 751876 478368 > 32⁵⁵ [i]