MinT - Best Known (28, 45, s)-Nets in Base 32

Best Known (28, 45, s)-Nets in Base 32

(28, 45, 330)-Net over F₃₂ — Constructive and digital

Digital (28, 45, 330)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 33)-net over F₃₂, using
  - s-reduction based on digital (0, 1, s)-net over F₃₂ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (0, 1, 33)-net over F₃₂ (see above)
3. digital (0, 2, 33)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
4. digital (0, 2, 33)-net over F₃₂ (see above)
5. digital (0, 2, 33)-net over F₃₂ (see above)
6. 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]
7. digital (0, 4, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
8. digital (0, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
9. digital (0, 8, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
10. digital (0, 17, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(28, 45, 516)-Net in Base 32 — Constructive

(28, 45, 516)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (7, 15, 258)-net in base 32, using
  - 1 times m-reduction [i] based on (7, 16, 258)-net in base 32, using
    - base change [i] based on 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. (13, 30, 258)-net in base 32, using
  - 2 times m-reduction [i] based on (13, 32, 258)-net in base 32, using
    - base change [i] based on digital (1, 20, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(28, 45, 3761)-Net over F₃₂ — Digital

Digital (28, 45, 3761)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(28, 45, large)-Net in Base 32 — Upper bound on s

There is no (28, 45, large)-net in base 32, because

15 times m-reduction [i] would yield (28, 30, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]