MinT - Best Known (42−17, 42, s)-Nets in Base 32

Best Known (42−17, 42, s)-Nets in Base 32

(42−17, 42, 242)-Net over F₃₂ — Constructive and digital

Digital (25, 42, 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, 2, 33)-net over F₃₂ (see above)
3. 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]
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, 5, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
6. digital (0, 8, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
7. digital (1, 18, 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]

(42−17, 42, 515)-Net in Base 32 — Constructive

(25, 42, 515)-net in base 32, using

base change [i] based on (18, 35, 515)-net in base 64, using
- (u,Â u+v)-construction [i] based on
  1. (3, 11, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (3, 12, 257)-net in base 64, using
      - base change [i] based on digital (0, 9, 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. (7, 24, 258)-net in base 64, using
    - base change [i] based on digital (1, 18, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(42−17, 42, 1968)-Net over F₃₂ — Digital

Digital (25, 42, 1968)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(42−17, 42, 6283791)-Net in Base 32 — Upper bound on s

There is no (25, 42, 6283792)-net in base 32, because

1 times m-reduction [i] would yield (25, 41, 6283792)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 51 422065 022158 123709 217964 306606 604260 871385 498108 585784 861795 > 32⁴¹ [i]