MinT - Best Known (89−55, 89, s)-Nets in Base 32

Best Known (89−55, 89, s)-Nets in Base 32

(89−55, 89, 131)-Net over F₃₂ — Constructive and digital

Digital (34, 89, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 27, 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 (7, 62, 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]

(89−55, 89, 257)-Net in Base 32 — Constructive

(34, 89, 257)-net in base 32, using

1 times m-reduction [i] based on (34, 90, 257)-net in base 32, using
- base change [i] based on (19, 75, 257)-net in base 64, using
  - 1 times m-reduction [i] based on (19, 76, 257)-net in base 64, using
    - base change [i] based on digital (0, 57, 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]

(89−55, 89, 273)-Net over F₃₂ — Digital

Digital (34, 89, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 89, 273)-net over F₃₂, using
- net from sequence [i] based on digital (30, 272)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 30 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(89−55, 89, 315)-Net in Base 32

(34, 89, 315)-net in base 32, using

1 times m-reduction [i] based on (34, 90, 315)-net in base 32, using
- base change [i] based on digital (19, 75, 315)-net over F₆₄, using
  - net from sequence [i] based on digital (19, 314)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 19 and N(F) ≥ 315, using
      - a curve from the manYPoints database [i]

(89−55, 89, 28346)-Net in Base 32 — Upper bound on s

There is no (34, 89, 28347)-net in base 32, because

1 times m-reduction [i] would yield (34, 88, 28347)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 839295 260730 517195 454067 710224 744021 730205 747160 994951 675845 649845 152809 866084 881753 380376 164646 705761 766339 723317 667803 398525 873280 > 32⁸⁸ [i]