MinT - Best Known (57, 57+41, s)-Nets in Base 32

Best Known (57, 57+41, s)-Nets in Base 32

(57, 57+41, 300)-Net over F₃₂ — Constructive and digital

Digital (57, 98, 300)-net over F₃₂, using

32¹ times duplication [i] based on digital (56, 97, 300)-net over F₃₂, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 20, 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 (7, 27, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)
  3. digital (9, 50, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(57, 57+41, 514)-Net in Base 32 — Constructive

(57, 98, 514)-net in base 32, using

(u,Â u+v)-construction [i] based on
1. (12, 32, 257)-net in base 32, using
  - base change [i] based on digital (0, 20, 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]
2. (25, 66, 257)-net in base 32, using
  - base change [i] based on (14, 55, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (14, 56, 257)-net in base 64, using
      - base change [i] based on digital (0, 42, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(57, 57+41, 2498)-Net over F₃₂ — Digital

Digital (57, 98, 2498)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+41, 5344672)-Net in Base 32 — Upper bound on s

There is no (57, 98, 5344673)-net in base 32, because

1 times m-reduction [i] would yield (57, 97, 5344673)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 99 896099 350782 866499 492300 544870 283496 107931 056244 051951 267108 484171 107578 185311 338754 516304 519915 365354 486496 802418 824600 587003 756982 198681 047016 > 32⁹⁷ [i]