MinT - Best Known (65, 65+42, s)-Nets in Base 32

Best Known (65, 65+42, s)-Nets in Base 32

(65, 65+42, 338)-Net over F₃₂ — Constructive and digital

Digital (65, 107, 338)-net over F₃₂, using

2 times m-reduction [i] based on digital (65, 109, 338)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 21, 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 (11, 33, 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, 55, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂ (see above)

(65, 65+42, 557)-Net in Base 32 — Constructive

(65, 107, 557)-net in base 32, using

1 times m-reduction [i] based on (65, 108, 557)-net in base 32, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 22, 44)-net over F₃₂, using
    - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
        a function field by SÃ©mirat [i]
  2. (43, 86, 513)-net in base 32, using
    - 4 times m-reduction [i] based on (43, 90, 513)-net in base 32, using
      - base change [i] based on digital (28, 75, 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+42, 4432)-Net over F₃₂ — Digital

Digital (65, 107, 4432)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+42, large)-Net in Base 32 — Upper bound on s

There is no (65, 107, large)-net in base 32, because

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