MinT - Best Known (21, 21+30, s)-Nets in Base 32

Best Known (21, 21+30, s)-Nets in Base 32

(21, 21+30, 128)-Net over F₃₂ — Constructive and digital

Digital (21, 51, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 18, 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, 33, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(21, 21+30, 185)-Net over F₃₂ — Digital

Digital (21, 51, 185)-net over F₃₂, using

net from sequence [i] based on digital (21, 184)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 21 and N(F) ≥ 185, using
  - a curve from the manYPoints database [i]

(21, 21+30, 257)-Net in Base 32 — Constructive

(21, 51, 257)-net in base 32, using

5 times m-reduction [i] based on (21, 56, 257)-net in base 32, using
- base change [i] based on digital (0, 35, 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]

(21, 21+30, 27151)-Net in Base 32 — Upper bound on s

There is no (21, 51, 27152)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 57909 307398 982216 256711 068795 290772 554631 136475 710340 494300 446977 980803 297983 > 32⁵¹ [i]