MinT - Best Known (54−29, 54, s)-Nets in Base 32

Best Known (54−29, 54, s)-Nets in Base 32

(54−29, 54, 162)-Net over F₃₂ — Constructive and digital

Digital (25, 54, 162)-net over F₃₂, using

1 times m-reduction [i] based on digital (25, 55, 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, 37, 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]

(54−29, 54, 288)-Net in Base 32 — Constructive

(25, 54, 288)-net in base 32, using

2 times m-reduction [i] based on (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]

(54−29, 54, 305)-Net over F₃₂ — Digital

Digital (25, 54, 305)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−29, 54, 321)-Net in Base 32

(25, 54, 321)-net in base 32, using

6 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]

(54−29, 54, 97304)-Net in Base 32 — Upper bound on s

There is no (25, 54, 97305)-net in base 32, because

1 times m-reduction [i] would yield (25, 53, 97305)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 59 290062 709326 239177 897991 915458 666411 402784 933916 529627 750679 620084 815896 061224 > 32⁵³ [i]