MinT - Best Known (109−64, 109, s)-Nets in Base 32

Best Known (109−64, 109, s)-Nets in Base 32

(109−64, 109, 174)-Net over F₃₂ — Constructive and digital

Digital (45, 109, 174)-net over F₃₂, using

t-expansion [i] based on digital (44, 109, 174)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 37, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]
  2. digital (7, 72, 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]

(109−64, 109, 288)-Net in Base 32 — Constructive

(45, 109, 288)-net in base 32, using

32¹ times duplication [i] based on (44, 108, 288)-net in base 32, using
- t-expansion [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]

(109−64, 109, 325)-Net over F₃₂ — Digital

Digital (45, 109, 325)-net over F₃₂, using

t-expansion [i] based on digital (44, 109, 325)-net over F₃₂, using
- net from sequence [i] based on digital (44, 324)-sequence over F₃₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 44 and N(F) ≥ 325, using
    - a curve from the manYPoints database [i]

(109−64, 109, 425)-Net in Base 32

(45, 109, 425)-net in base 32, using

32¹ times duplication [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]

(109−64, 109, 55246)-Net in Base 32 — Upper bound on s

There is no (45, 109, 55247)-net in base 32, because

the generalized Rao bound for nets shows that 32^mÂ â‰¥ 115 196337 030582 790414 436895 485105 500028 018238 749213 155614 790052 574750 565023 558807 296419 859198 008119 114207 315331 275868 427455 617845 152919 427400 075023 439619 785739 733138 > 32¹⁰⁹ [i]