MinT - Best Known (109−68, 109, s)-Nets in Base 32

Best Known (109−68, 109, s)-Nets in Base 32

(109−68, 109, 131)-Net over F₃₂ — Constructive and digital

Digital (41, 109, 131)-net over F₃₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 34, 33)-net over F₃₂, using
  - net from sequence [i] based on digital (0, 32)-sequence over F₃₂, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 0 and N(F) ≥ 33, using
      - the rational function field F₃₂(x) [i]
    - Niederreiter sequence [i]
2. digital (7, 75, 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]

(109−68, 109, 288)-Net in Base 32 — Constructive

(41, 109, 288)-net in base 32, using

32¹ times duplication [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]

(109−68, 109, 308)-Net over F₃₂ — Digital

Digital (41, 109, 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]

(109−68, 109, 342)-Net in Base 32

(41, 109, 342)-net in base 32, using

32¹ times duplication [i] based on (40, 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]

(109−68, 109, 29186)-Net in Base 32 — Upper bound on s

There is no (41, 109, 29187)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 205474 658751 330561 878115 374393 219361 149280 849575 716642 975210 312311 905265 840575 018843 670279 000580 261563 004973 213166 515923 852644 949268 020325 555635 114287 263253 816688 > 32¹⁰⁹ [i]