MinT - Best Known (33, 53, s)-Nets in Base 32

Best Known (33, 53, s)-Nets in Base 32

(33, 53, 275)-Net over F₃₂ — Constructive and digital

Digital (33, 53, 275)-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 (1, 21, 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]

(33, 53, 516)-Net in Base 32 — Constructive

(33, 53, 516)-net in base 32, using

1 times m-reduction [i] based on (33, 54, 516)-net in base 32, using
- base change [i] based on (24, 45, 516)-net in base 64, using
  - 64¹ times duplication [i] based on (23, 44, 516)-net in base 64, using
    - base change [i] based on digital (12, 33, 516)-net over F₂₅₆, using
      - (u,Â u+v)-construction [i] based on
        digital (1, 11, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
        digital (1, 22, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(33, 53, 4050)-Net over F₃₂ — Digital

Digital (33, 53, 4050)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(33, 53, large)-Net in Base 32 — Upper bound on s

There is no (33, 53, large)-net in base 32, because

18 times m-reduction [i] would yield (33, 35, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]