MinT - Best Known (68, 68+40, s)-Nets in Base 32

Best Known (68, 68+40, s)-Nets in Base 32

(68, 68+40, 360)-Net over F₃₂ — Constructive and digital

Digital (68, 108, 360)-net over F₃₂, using

2 times m-reduction [i] based on digital (68, 110, 360)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (11, 25, 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]
  2. digital (11, 32, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)
  3. digital (11, 53, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(68, 68+40, 593)-Net in Base 32 — Constructive

(68, 108, 593)-net in base 32, using

t-expansion [i] based on (67, 108, 593)-net in base 32, using
- base change [i] based on digital (49, 90, 593)-net over F₆₄, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 21, 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. digital (28, 69, 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]

(68, 68+40, 7334)-Net over F₃₂ — Digital

Digital (68, 108, 7334)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+40, large)-Net in Base 32 — Upper bound on s

There is no (68, 108, large)-net in base 32, because

38 times m-reduction [i] would yield (68, 70, large)-net in base 32, but
- mutually orthogonal hypercube bound [i]