MinT - Best Known (40−16, 40, s)-Nets in Base 32

Best Known (40−16, 40, s)-Nets in Base 32

(40−16, 40, 231)-Net over F₃₂ — Constructive and digital

Digital (24, 40, 231)-net over F₃₂, using

1 times m-reduction [i] based on digital (24, 41, 231)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 2, 33)-net over F₃₂, using
    - kÂ =Â 2 construction [i]
  2. digital (0, 2, 33)-net over F₃₂ (see above)
  3. digital (0, 3, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
        the rational function field F₃₂(x) [i]
      - Niederreiter sequence [i]
  4. digital (0, 4, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  5. digital (0, 5, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  6. digital (0, 8, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
  7. digital (0, 17, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(40−16, 40, 515)-Net in Base 32 — Constructive

(24, 40, 515)-net in base 32, using

base change [i] based on digital (9, 25, 515)-net over F₂₅₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (0, 8, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]
  2. digital (1, 17, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(40−16, 40, 2146)-Net over F₃₂ — Digital

Digital (24, 40, 2146)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(40−16, 40, 4074533)-Net in Base 32 — Upper bound on s

There is no (24, 40, 4074534)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1 606940 515890 568777 169964 027432 304994 911931 445617 631402 696724 > 32⁴⁰ [i]