MinT - Best Known (27, 27+16, s)-Nets in Base 32

Best Known (27, 27+16, s)-Nets in Base 32

(27, 27+16, 297)-Net over F₃₂ — Constructive and digital

Digital (27, 43, 297)-net over F₃₂, using

1 times m-reduction [i] based on 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+16, 516)-Net in Base 32 — Constructive

(27, 43, 516)-net in base 32, using

32¹ times duplication [i] based on (26, 42, 516)-net in base 32, using
- base change [i] based on (19, 35, 516)-net in base 64, using
  - 1 times m-reduction [i] based on (19, 36, 516)-net in base 64, using
    - base change [i] based on digital (10, 27, 516)-net over F₂₅₆, using
      - (u,Â u+v)-construction [i] based on
        digital (1, 9, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
        digital (1, 18, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(27, 27+16, 4285)-Net over F₃₂ — Digital

Digital (27, 43, 4285)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(27, 27+16, large)-Net in Base 32 — Upper bound on s

There is no (27, 43, large)-net in base 32, because

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