MinT - Best Known (47−19, 47, s)-Nets in Base 32

Best Known (47−19, 47, s)-Nets in Base 32

(47−19, 47, 242)-Net over F₃₂ — Constructive and digital

Digital (28, 47, 242)-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, 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]
3. digital (0, 3, 33)-net over F₃₂ (see above)
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, 6, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
6. digital (0, 9, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
7. digital (1, 20, 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]

(47−19, 47, 515)-Net in Base 32 — Constructive

(28, 47, 515)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (6, 15, 257)-net in base 32, using
  - 1 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, 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]

(47−19, 47, 2083)-Net over F₃₂ — Digital

Digital (28, 47, 2083)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(47−19, 47, 6597482)-Net in Base 32 — Upper bound on s

There is no (28, 47, 6597483)-net in base 32, because

1 times m-reduction [i] would yield (28, 46, 6597483)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 1725 437656 072894 337003 941334 491104 501403 790935 204724 518861 085958 670862 > 32⁴⁶ [i]