MinT - Best Known (81−48, 81, s)-Nets in Base 32

Best Known (81−48, 81, s)-Nets in Base 32

(81−48, 81, 142)-Net over F₃₂ — Constructive and digital

Digital (33, 81, 142)-net over F₃₂, using

2 times m-reduction [i] based on digital (33, 83, 142)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 26, 44)-net over F₃₂, using
    - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
        a function field by SÃ©mirat [i]
  2. digital (7, 57, 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]

(81−48, 81, 273)-Net over F₃₂ — Digital

Digital (33, 81, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 81, 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]

(81−48, 81, 288)-Net in Base 32 — Constructive

(33, 81, 288)-net in base 32, using

3 times m-reduction [i] based on (33, 84, 288)-net in base 32, using
- base change [i] based on digital (9, 60, 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]

(81−48, 81, 321)-Net in Base 32

(33, 81, 321)-net in base 32, using

32¹ times duplication [i] based on (32, 80, 321)-net in base 32, using
- base change [i] based on digital (2, 50, 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]

(81−48, 81, 37996)-Net in Base 32 — Upper bound on s

There is no (33, 81, 37997)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 82 665554 133736 522739 444287 633769 266054 643245 179563 975607 545424 567764 790828 132631 016767 882830 223203 071958 351834 093399 770818 > 32⁸¹ [i]