MinT - Best Known (65, 65+38, s)-Nets in Base 32

Best Known (65, 65+38, s)-Nets in Base 32

(65, 65+38, 360)-Net over F₃₂ — Constructive and digital

Digital (65, 103, 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, 19, 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, 45, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(65, 65+38, 578)-Net in Base 32 — Constructive

(65, 103, 578)-net in base 32, using

1 times m-reduction [i] based on (65, 104, 578)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. (4, 23, 65)-net in base 32, using
    - 1 times m-reduction [i] based on (4, 24, 65)-net in base 32, using
      - base change [i] based on digital (0, 20, 65)-net over F₆₄, using
        net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [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]

(65, 65+38, 7339)-Net over F₃₂ — Digital

Digital (65, 103, 7339)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+38, large)-Net in Base 32 — Upper bound on s

There is no (65, 103, large)-net in base 32, because

36 times m-reduction [i] would yield (65, 67, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]