MinT - Best Known (45, 45+59, s)-Nets in Base 32

Best Known (45, 45+59, s)-Nets in Base 32

(45, 45+59, 202)-Net over F₃₂ — Constructive and digital

Digital (45, 104, 202)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 36, 98)-net over F₃₂, using
  - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
      - a function field by SÃ©mirat [i]
2. digital (9, 68, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(45, 45+59, 288)-Net in Base 32 — Constructive

(45, 104, 288)-net in base 32, using

t-expansion [i] based on (40, 104, 288)-net in base 32, using
- 4 times m-reduction [i] based on (40, 108, 288)-net in base 32, using
  - base change [i] based on (22, 90, 288)-net in base 64, using
    - 1 times m-reduction [i] based on (22, 91, 288)-net in base 64, using
      - base change [i] based on digital (9, 78, 288)-net over F₁₂₈, using
        net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(45, 45+59, 352)-Net over F₃₂ — Digital

Digital (45, 104, 352)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+59, 425)-Net in Base 32

(45, 104, 425)-net in base 32, using

t-expansion [i] based on (44, 104, 425)-net in base 32, using
- 4 times m-reduction [i] based on (44, 108, 425)-net in base 32, using
  - base change [i] based on digital (26, 90, 425)-net over F₆₄, using
    - net from sequence [i] based on digital (26, 424)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 26 and N(F) ≥ 425, using
        a curve from the manYPoints database [i]

(45, 45+59, 83475)-Net in Base 32 — Upper bound on s

There is no (45, 104, 83476)-net in base 32, because

1 times m-reduction [i] would yield (45, 103, 83476)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 107273 068018 939904 846008 532418 457646 576601 146899 083329 854495 551680 391785 957465 593145 105752 378682 289089 470466 100482 651837 017935 803256 323925 644349 879733 389696 > 32¹⁰³ [i]