MinT - Best Known (75−30, 75, s)-Nets in Base 32

Best Known (75−30, 75, s)-Nets in Base 32

(75−30, 75, 273)-Net over F₃₂ — Constructive and digital

Digital (45, 75, 273)-net over F₃₂, using

1 times m-reduction [i] based on digital (45, 76, 273)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (6, 16, 77)-net over F₃₂, using
    - (u,Â u+v)-construction [i] based on
      1. digital (0, 5, 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 (1, 11, 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. digital (7, 22, 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, 38, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(75−30, 75, 514)-Net in Base 32 — Constructive

(45, 75, 514)-net in base 32, using

2 times m-reduction [i] based on (45, 77, 514)-net in base 32, using
- base change [i] based on (23, 55, 514)-net in base 128, using
  - 1 times m-reduction [i] based on (23, 56, 514)-net in base 128, using
    - base change [i] based on digital (16, 49, 514)-net over F₂₅₆, using
      - (u,Â u+v)-construction [i] based on
        digital (0, 16, 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]
        digital (0, 33, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(75−30, 75, 2955)-Net over F₃₂ — Digital

Digital (45, 75, 2955)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(75−30, 75, 6952714)-Net in Base 32 — Upper bound on s

There is no (45, 75, 6952715)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 76957 193455 513481 299110 802158 397118 273498 258105 613776 563988 388312 597767 186664 837123 269457 209508 736661 652481 009408 > 32⁷⁵ [i]