MinT - Best Known (27, 45, s)-Nets in Base 32

Best Known (27, 45, s)-Nets in Base 32

(27, 45, 231)-Net over F₃₂ — Constructive and digital

Digital (27, 45, 231)-net over F₃₂, using

1 times m-reduction [i] based on digital (27, 46, 231)-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 (0, 19, 33)-net over F₃₂, using
    - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)

(27, 45, 514)-Net in Base 32 — Constructive

(27, 45, 514)-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. (12, 30, 257)-net in base 32, using
  - 2 times m-reduction [i] based on (12, 32, 257)-net in base 32, using
    - base change [i] based on digital (0, 20, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(27, 45, 2241)-Net over F₃₂ — Digital

Digital (27, 45, 2241)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 45, 4488892)-Net in Base 32 — Upper bound on s

There is no (27, 45, 4488893)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 53 919918 627829 167283 294711 939684 825680 130515 488035 524073 347272 686940 > 32⁴⁵ [i]