MinT - Best Known (54, 54+32, s)-Nets in Base 32

Best Known (54, 54+32, s)-Nets in Base 32

(54, 54+32, 324)-Net over F₃₂ — Constructive and digital

Digital (54, 86, 324)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 11, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (3, 13, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)
3. digital (7, 23, 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]
4. digital (7, 39, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(54, 54+32, 545)-Net in Base 32 — Constructive

(54, 86, 545)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (11, 27, 257)-net in base 32, using
  - 1 times m-reduction [i] based on (11, 28, 257)-net in base 32, using
    - base change [i] based on (3, 20, 257)-net in base 128, using
      - 4 times m-reduction [i] based on (3, 24, 257)-net in base 128, using
        base change [i] based on digital (0, 21, 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. (27, 59, 288)-net in base 32, using
  - 4 times m-reduction [i] based on (27, 63, 288)-net in base 32, using
    - base change [i] based on digital (9, 45, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(54, 54+32, 6017)-Net over F₃₂ — Digital

Digital (54, 86, 6017)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+32, large)-Net in Base 32 — Upper bound on s

There is no (54, 86, large)-net in base 32, because

30 times m-reduction [i] would yield (54, 56, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]