MinT - Best Known (56, 56+32, s)-Nets in Base 32

Best Known (56, 56+32, s)-Nets in Base 32

(56, 56+32, 339)-Net over F₃₂ — Constructive and digital

Digital (56, 88, 339)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (8, 18, 99)-net over F₃₂, using
  - 1 times m-reduction [i] based on digital (8, 19, 99)-net over F₃₂, using
    - generalized (u,Â u+v)-construction [i] based on
      1. 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]
      2. digital (0, 5, 33)-net over F₃₂, using
        net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
      3. digital (0, 11, 33)-net over F₃₂, using
        net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
2. digital (11, 27, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]
3. digital (11, 43, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(56, 56+32, 546)-Net in Base 32 — Constructive

(56, 88, 546)-net in base 32, using

2 times m-reduction [i] based on (56, 90, 546)-net in base 32, using
- base change [i] based on (41, 75, 546)-net in base 64, using
  - (u,Â u+v)-construction [i] based on
    1. (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]
    2. (17, 51, 288)-net in base 64, using
      - 5 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
        base change [i] based on digital (9, 48, 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]

(56, 56+32, 7521)-Net over F₃₂ — Digital

Digital (56, 88, 7521)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+32, large)-Net in Base 32 — Upper bound on s

There is no (56, 88, large)-net in base 32, because

30 times m-reduction [i] would yield (56, 58, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]