MinT - Best Known (31, 50, s)-Nets in Base 32

Best Known (31, 50, s)-Nets in Base 32

(31, 50, 297)-Net over F₃₂ — Constructive and digital

Digital (31, 50, 297)-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, 2, 33)-net over F₃₂ (see above)
4. 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]
5. digital (0, 3, 33)-net over F₃₂ (see above)
6. digital (0, 4, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
7. digital (0, 6, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
8. digital (0, 9, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
9. digital (0, 19, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(31, 50, 516)-Net in Base 32 — Constructive

(31, 50, 516)-net in base 32, using

32² times duplication [i] based on (29, 48, 516)-net in base 32, using
- base change [i] based on digital (11, 30, 516)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 10, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
    2. digital (1, 20, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(31, 50, 3705)-Net over F₃₂ — Digital

Digital (31, 50, 3705)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(31, 50, large)-Net in Base 32 — Upper bound on s

There is no (31, 50, large)-net in base 32, because

17 times m-reduction [i] would yield (31, 33, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]