MinT - Best Known (57−21, 57, s)-Nets in Base 32

Best Known (57−21, 57, s)-Nets in Base 32

(57−21, 57, 297)-Net over F₃₂ — Constructive and digital

Digital (36, 57, 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, 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, 3, 33)-net over F₃₂ (see above)
5. digital (0, 4, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
6. digital (0, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
7. digital (0, 7, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
8. digital (0, 10, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
9. digital (0, 21, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(57−21, 57, 518)-Net in Base 32 — Constructive

(36, 57, 518)-net in base 32, using

32¹ times duplication [i] based on (35, 56, 518)-net in base 32, using
- base change [i] based on digital (14, 35, 518)-net over F₂₅₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (2, 12, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
    2. digital (2, 23, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆ (see above)

(57−21, 57, 5229)-Net over F₃₂ — Digital

Digital (36, 57, 5229)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−21, 57, large)-Net in Base 32 — Upper bound on s

There is no (36, 57, large)-net in base 32, because

19 times m-reduction [i] would yield (36, 38, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]