MinT - Best Known (32, 32+52, s)-Nets in Base 32

Best Known (32, 32+52, s)-Nets in Base 32

(32, 32+52, 128)-Net over F₃₂ — Constructive and digital

Digital (32, 84, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 29, 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, 55, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(32, 32+52, 257)-Net in Base 32 — Constructive

(32, 84, 257)-net in base 32, using

base change [i] based on (18, 70, 257)-net in base 64, using
- 2 times m-reduction [i] based on (18, 72, 257)-net in base 64, using
  - base change [i] based on digital (0, 54, 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]

(32, 32+52, 273)-Net over F₃₂ — Digital

Digital (32, 84, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 84, 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]

(32, 32+52, 281)-Net in Base 32

(32, 84, 281)-net in base 32, using

base change [i] based on digital (18, 70, 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]

(32, 32+52, 24802)-Net in Base 32 — Upper bound on s

There is no (32, 84, 24803)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 709971 403332 625142 024981 160717 168573 807534 194881 353899 880453 819240 779266 686495 355528 317801 107568 794774 821816 722574 974690 295556 > 32⁸⁴ [i]