MinT - Best Known (52−20, 52, s)-Nets in Base 32

Best Known (52−20, 52, s)-Nets in Base 32

(52−20, 52, 264)-Net over F₃₂ — Constructive and digital

Digital (32, 52, 264)-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, 6, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
7. digital (0, 10, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
8. digital (0, 20, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(52−20, 52, 516)-Net in Base 32 — Constructive

(32, 52, 516)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (6, 16, 257)-net in base 32, using
  - base change [i] based on digital (0, 10, 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. (16, 36, 259)-net in base 32, using
  - base change [i] based on (10, 30, 259)-net in base 64, using
    - 2 times m-reduction [i] based on (10, 32, 259)-net in base 64, using
      - base change [i] based on digital (2, 24, 259)-net over F₂₅₆, using
        net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(52−20, 52, 3376)-Net over F₃₂ — Digital

Digital (32, 52, 3376)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(52−20, 52, large)-Net in Base 32 — Upper bound on s

There is no (32, 52, large)-net in base 32, because

18 times m-reduction [i] would yield (32, 34, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]