MinT - Best Known (42, 85, s)-Nets in Base 32

Best Known (42, 85, s)-Nets in Base 32

(42, 85, 224)-Net over F₃₂ — Constructive and digital

Digital (42, 85, 224)-net over F₃₂, using

1 times m-reduction [i] based on digital (42, 86, 224)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 31, 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]
  2. digital (11, 55, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(42, 85, 288)-Net in Base 32 — Constructive

(42, 85, 288)-net in base 32, using

t-expansion [i] based on (40, 85, 288)-net in base 32, using
- 23 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]

(42, 85, 613)-Net over F₃₂ — Digital

Digital (42, 85, 613)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 85, 293572)-Net in Base 32 — Upper bound on s

There is no (42, 85, 293573)-net in base 32, because

1 times m-reduction [i] would yield (42, 84, 293573)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 2 707831 805721 371766 778802 959862 976474 861446 826133 373440 248674 347292 868656 964580 511483 481410 175319 685135 930891 272668 890666 373824 > 32⁸⁴ [i]