MinT - Best Known (41, 41+64, s)-Nets in Base 32

Best Known (41, 41+64, s)-Nets in Base 32

(41, 41+64, 142)-Net over F₃₂ — Constructive and digital

Digital (41, 105, 142)-net over F₃₂, using

2 times m-reduction [i] based on digital (41, 107, 142)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (1, 34, 44)-net over F₃₂, using
    - net from sequence [i] based on digital (1, 43)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 1 and N(F) ≥ 44, using
        a function field by SÃ©mirat [i]
  2. digital (7, 73, 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+64, 288)-Net in Base 32 — Constructive

(41, 105, 288)-net in base 32, using

t-expansion [i] based on (40, 105, 288)-net in base 32, using
- 3 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+64, 308)-Net over F₃₂ — Digital

Digital (41, 105, 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+64, 342)-Net in Base 32

(41, 105, 342)-net in base 32, using

t-expansion [i] based on (38, 105, 342)-net in base 32, using
- 3 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+64, 35817)-Net in Base 32 — Upper bound on s

There is no (41, 105, 35818)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 915135 066777 771367 690945 976786 657980 079976 503122 856037 424256 611356 153297 719605 189483 066094 485881 136106 358950 076330 401178 667739 231821 628581 809866 512977 739975 > 32¹⁰⁵ [i]