MinT - Best Known (41, 41+63, s)-Nets in Base 32

Best Known (41, 41+63, s)-Nets in Base 32

(41, 41+63, 162)-Net over F₃₂ — Constructive and digital

Digital (41, 104, 162)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 34, 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 (7, 70, 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]

(41, 41+63, 288)-Net in Base 32 — Constructive

(41, 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]

(41, 41+63, 308)-Net over F₃₂ — Digital

Digital (41, 104, 308)-net over F₃₂, using

net from sequence [i] based on digital (41, 307)-sequence over F₃₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 41 and N(F) ≥ 308, using
  - a curve from the manYPoints database [i]

(41, 41+63, 342)-Net in Base 32

(41, 104, 342)-net in base 32, using

t-expansion [i] based on (38, 104, 342)-net in base 32, using
- 4 times m-reduction [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]

(41, 41+63, 40131)-Net in Base 32 — Upper bound on s

There is no (41, 104, 40132)-net in base 32, because

1 times m-reduction [i] would yield (41, 103, 40132)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 107342 487828 503912 951587 577509 121245 163688 590700 265520 430888 989641 651941 511995 453048 756165 642589 748661 445581 626839 262938 704054 490202 628055 161900 647903 953008 > 32¹⁰³ [i]