MinT - Best Known (34, 34+57, s)-Nets in Base 32

Best Known (34, 34+57, s)-Nets in Base 32

(34, 34+57, 128)-Net over F₃₂ — Constructive and digital

Digital (34, 91, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 31, 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, 60, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(34, 34+57, 216)-Net in Base 32 — Constructive

(34, 91, 216)-net in base 32, using

t-expansion [i] based on (33, 91, 216)-net in base 32, using
- 7 times m-reduction [i] based on (33, 98, 216)-net in base 32, using
  - base change [i] based on digital (5, 70, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(34, 34+57, 273)-Net over F₃₂ — Digital

Digital (34, 91, 273)-net over F₃₂, using

t-expansion [i] based on digital (30, 91, 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]

(34, 34+57, 281)-Net in Base 32

(34, 91, 281)-net in base 32, using

5 times m-reduction [i] based on (34, 96, 281)-net in base 32, using
- base change [i] based on digital (18, 80, 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]

(34, 34+57, 25082)-Net in Base 32 — Upper bound on s

There is no (34, 91, 25083)-net in base 32, because

1 times m-reduction [i] would yield (34, 90, 25083)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2907 982605 969043 230671 301452 563011 508184 020985 201388 937980 657232 325868 894823 837564 506382 991748 459570 365413 241151 219148 701862 457028 326040 > 32⁹⁰ [i]