MinT - Best Known (76−47, 76, s)-Nets in Base 32

Best Known (76−47, 76, s)-Nets in Base 32

(76−47, 76, 128)-Net over F₃₂ — Constructive and digital

Digital (29, 76, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 26, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (3, 50, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(76−47, 76, 257)-Net in Base 32 — Constructive

(29, 76, 257)-net in base 32, using

1 times m-reduction [i] based on (29, 77, 257)-net in base 32, using
- base change [i] based on (7, 55, 257)-net in base 128, using
  - 1 times m-reduction [i] based on (7, 56, 257)-net in base 128, using
    - base change [i] based on digital (0, 49, 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]

(76−47, 76, 257)-Net over F₃₂ — Digital

Digital (29, 76, 257)-net over F₃₂, using

t-expansion [i] based on digital (28, 76, 257)-net over F₃₂, using
- net from sequence [i] based on digital (28, 256)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 28 and N(F) ≥ 257, using
    - a curve from the manYPoints database [i]

(76−47, 76, 267)-Net in Base 32

(29, 76, 267)-net in base 32, using

2 times m-reduction [i] based on (29, 78, 267)-net in base 32, using
- base change [i] based on digital (16, 65, 267)-net over F₆₄, using
  - net from sequence [i] based on digital (16, 266)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 16 and N(F) ≥ 267, using
      - a curve from the manYPoints database [i]

(76−47, 76, 24602)-Net in Base 32 — Upper bound on s

There is no (29, 76, 24603)-net in base 32, because

1 times m-reduction [i] would yield (29, 75, 24603)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 76968 961606 173017 176393 430447 751746 251484 525885 920962 830100 378520 902278 968806 449642 535792 572998 256400 720149 036568 > 32⁷⁵ [i]