MinT - Best Known (87−54, 87, s)-Nets in Base 32

Best Known (87−54, 87, s)-Nets in Base 32

(87−54, 87, 128)-Net over F₃₂ — Constructive and digital

Digital (33, 87, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 30, 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, 57, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(87−54, 87, 257)-Net in Base 32 — Constructive

(33, 87, 257)-net in base 32, using

1 times m-reduction [i] based on (33, 88, 257)-net in base 32, using
- base change [i] based on digital (0, 55, 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]

(87−54, 87, 273)-Net over F₃₂ — Digital

Digital (33, 87, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 87, 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]

(87−54, 87, 281)-Net in Base 32

(33, 87, 281)-net in base 32, using

3 times m-reduction [i] based on (33, 90, 281)-net in base 32, using
- base change [i] based on digital (18, 75, 281)-net over F₆₄, using
  - net from sequence [i] based on digital (18, 280)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 18 and N(F) ≥ 281, using
      - a curve from the manYPoints database [i]

(87−54, 87, 24930)-Net in Base 32 — Upper bound on s

There is no (33, 87, 24931)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 88772 043963 968582 398232 492512 436475 444027 439229 975892 638630 045417 860652 380571 541235 575681 669130 084931 839717 310742 698117 526538 415776 > 32⁸⁷ [i]