MinT - Best Known (72−27, 72, s)-Nets in Base 32

Best Known (72−27, 72, s)-Nets in Base 32

(72−27, 72, 300)-Net over F₃₂ — Constructive and digital

Digital (45, 72, 300)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 16, 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]
2. digital (7, 20, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
3. digital (9, 36, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(72−27, 72, 545)-Net in Base 32 — Constructive

(45, 72, 545)-net in base 32, using

base change [i] based on (33, 60, 545)-net in base 64, using
- (u,Â u+v)-construction [i] based on
  1. (5, 18, 257)-net in base 64, using
    - 2 times m-reduction [i] based on (5, 20, 257)-net in base 64, using
      - base change [i] based on digital (0, 15, 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. (15, 42, 288)-net in base 64, using
    - base change [i] based on digital (9, 36, 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]

(72−27, 72, 5025)-Net over F₃₂ — Digital

Digital (45, 72, 5025)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−27, 72, large)-Net in Base 32 — Upper bound on s

There is no (45, 72, large)-net in base 32, because

25 times m-reduction [i] would yield (45, 47, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]