MinT - Best Known (27, 27+17, s)-Nets in Base 32

Best Known (27, 27+17, s)-Nets in Base 32

(27, 27+17, 297)-Net over F₃₂ — Constructive and digital

Digital (27, 44, 297)-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, 2, 33)-net over F₃₂, using
  - kÂ =Â 2 construction [i]
3. digital (0, 2, 33)-net over F₃₂ (see above)
4. digital (0, 2, 33)-net over F₃₂ (see above)
5. 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]
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, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
8. digital (0, 8, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
9. digital (0, 17, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(27, 27+17, 515)-Net in Base 32 — Constructive

(27, 44, 515)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (6, 14, 257)-net in base 32, using
  - 2 times m-reduction [i] based on (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. (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, using
        Niederreiter sequence [i]

(27, 27+17, 3030)-Net over F₃₂ — Digital

Digital (27, 44, 3030)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 27+17, large)-Net in Base 32 — Upper bound on s

There is no (27, 44, large)-net in base 32, because

15 times m-reduction [i] would yield (27, 29, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]