MinT - Best Known (68−27, 68, s)-Nets in Base 32

Best Known (68−27, 68, s)-Nets in Base 32

(68−27, 68, 273)-Net over F₃₂ — Constructive and digital

Digital (41, 68, 273)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (5, 14, 77)-net over F₃₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (0, 4, 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]
    2. digital (1, 10, 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]
2. digital (7, 20, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]
3. digital (7, 34, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(68−27, 68, 516)-Net in Base 32 — Constructive

(41, 68, 516)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (8, 21, 257)-net in base 32, using
  - base change [i] based on (2, 15, 257)-net in base 128, using
    - 1 times m-reduction [i] based on (2, 16, 257)-net in base 128, using
      - base change [i] based on digital (0, 14, 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. (20, 47, 259)-net in base 32, using
  - 1 times m-reduction [i] based on (20, 48, 259)-net in base 32, using
    - base change [i] based on digital (2, 30, 259)-net over F₂₅₆, using
      - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(68−27, 68, 2954)-Net over F₃₂ — Digital

Digital (41, 68, 2954)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−27, 68, large)-Net in Base 32 — Upper bound on s

There is no (41, 68, large)-net in base 32, because

25 times m-reduction [i] would yield (41, 43, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]