MinT - Best Known (109−69, 109, s)-Nets in Base 32

Best Known (109−69, 109, s)-Nets in Base 32

(109−69, 109, 128)-Net over F₃₂ — Constructive and digital

Digital (40, 109, 128)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 37, 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, 72, 64)-net over F₃₂, using
  - net from sequence [i] based on digital (3, 63)-sequence over F₃₂ (see above)

(109−69, 109, 216)-Net in Base 32 — Constructive

(40, 109, 216)-net in base 32, using

32¹ times duplication [i] based on (39, 108, 216)-net in base 32, using
- t-expansion [i] based on (36, 108, 216)-net in base 32, using
  - base change [i] based on (18, 90, 216)-net in base 64, using
    - 1 times m-reduction [i] based on (18, 91, 216)-net in base 64, using
      - base change [i] based on digital (5, 78, 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]

(109−69, 109, 293)-Net over F₃₂ — Digital

Digital (40, 109, 293)-net over F₃₂, using

net from sequence [i] based on digital (40, 292)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 40 and N(F) ≥ 293, using
  - a curve from the manYPoints database [i]

(109−69, 109, 342)-Net in Base 32

(40, 109, 342)-net in base 32, using

32¹ times duplication [i] based on (39, 108, 342)-net in base 32, using
- t-expansion [i] based on (38, 108, 342)-net in base 32, using
  - base change [i] based on digital (20, 90, 342)-net over F₆₄, using
    - net from sequence [i] based on digital (20, 341)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 20 and N(F) ≥ 342, using
        a curve from the manYPoints database [i]

(109−69, 109, 26356)-Net in Base 32 — Upper bound on s

There is no (40, 109, 26357)-net in base 32, because

1 times m-reduction [i] would yield (40, 108, 26357)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 3 601220 748093 718829 274213 530002 386037 714981 777847 971684 449816 543834 319402 108740 875550 161757 291566 904875 492668 062053 778642 032580 151544 134427 612477 282107 463020 630614 > 32¹⁰⁸ [i]