MinT - Best Known (37, 37+49, s)-Nets in Base 32

Best Known (37, 37+49, s)-Nets in Base 32

(37, 37+49, 174)-Net over F₃₂ — Constructive and digital

Digital (37, 86, 174)-net over F₃₂, using

1 times m-reduction [i] based on digital (37, 87, 174)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 30, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, 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]

(37, 37+49, 288)-Net in Base 32 — Constructive

(37, 86, 288)-net in base 32, using

12 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+49, 294)-Net over F₃₂ — Digital

Digital (37, 86, 294)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(37, 37+49, 342)-Net in Base 32

(37, 86, 342)-net in base 32, using

16 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+49, 67711)-Net in Base 32 — Upper bound on s

There is no (37, 86, 67712)-net in base 32, because

1 times m-reduction [i] would yield (37, 85, 67712)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 86 664159 854446 293428 159652 847178 305141 830743 114563 441682 378282 409644 776617 736903 321645 283723 019384 082520 753508 790755 584864 705177 > 32⁸⁵ [i]