MinT - Best Known (66, 66+39, s)-Nets in Base 32

Best Known (66, 66+39, s)-Nets in Base 32

(66, 66+39, 360)-Net over F₃₂ — Constructive and digital

Digital (66, 105, 360)-net over F₃₂, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 13, 66)-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 (0, 9, 33)-net over F₃₂, using
      - net from sequence [i] based on digital (0, 32)-sequence over F₃₂ (see above)
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, 26, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
4. digital (7, 46, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(66, 66+39, 593)-Net in Base 32 — Constructive

(66, 105, 593)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (5, 24, 80)-net in base 32, using
  - base change [i] based on digital (1, 20, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. (42, 81, 513)-net in base 32, using
  - 3 times m-reduction [i] based on (42, 84, 513)-net in base 32, using
    - base change [i] based on digital (28, 70, 513)-net over F₆₄, using
      - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
        the Hermitian function field over F₆₄ [i]

(66, 66+39, 7008)-Net over F₃₂ — Digital

Digital (66, 105, 7008)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(66, 66+39, large)-Net in Base 32 — Upper bound on s

There is no (66, 105, large)-net in base 32, because

37 times m-reduction [i] would yield (66, 68, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]