MinT - Best Known (37, 37+59, s)-Nets in Base 32

Best Known (37, 37+59, s)-Nets in Base 32

(37, 37+59, 142)-Net over F₃₂ — Constructive and digital

Digital (37, 96, 142)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 30, 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, 66, 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]

(37, 37+59, 273)-Net over F₃₂ — Digital

Digital (37, 96, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 96, 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]

(37, 37+59, 288)-Net in Base 32 — Constructive

(37, 96, 288)-net in base 32, using

2 times m-reduction [i] based on (37, 98, 288)-net in base 32, using
- base change [i] based on digital (9, 70, 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]

(37, 37+59, 342)-Net in Base 32

(37, 96, 342)-net in base 32, using

6 times m-reduction [i] based on (37, 102, 342)-net in base 32, using
- base change [i] based on digital (20, 85, 342)-net over F₆₄, using
  - net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
      - a curve from the manYPoints database [i]

(37, 37+59, 32079)-Net in Base 32 — Upper bound on s

There is no (37, 96, 32080)-net in base 32, because

1 times m-reduction [i] would yield (37, 95, 32080)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 97635 647623 954244 491775 706210 016567 829252 446273 263868 786092 541375 080200 695099 206145 490452 747486 946272 220187 459334 662600 628353 438795 499651 417340 > 32⁹⁵ [i]