MinT - Best Known (45, 45+61, s)-Nets in Base 32

Best Known (45, 45+61, s)-Nets in Base 32

(45, 45+61, 196)-Net over F₃₂ — Constructive and digital

Digital (45, 106, 196)-net over F₃₂, using

1 times m-reduction [i] based on digital (45, 107, 196)-net over F₃₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 38, 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]
  2. digital (7, 69, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂ (see above)

(45, 45+61, 288)-Net in Base 32 — Constructive

(45, 106, 288)-net in base 32, using

t-expansion [i] based on (40, 106, 288)-net in base 32, using
- 2 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]

(45, 45+61, 329)-Net over F₃₂ — Digital

Digital (45, 106, 329)-net over F₃₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+61, 425)-Net in Base 32

(45, 106, 425)-net in base 32, using

t-expansion [i] based on (44, 106, 425)-net in base 32, using
- 2 times m-reduction [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]

(45, 45+61, 72004)-Net in Base 32 — Upper bound on s

There is no (45, 106, 72005)-net in base 32, because

1 times m-reduction [i] would yield (45, 105, 72005)-net in base 32, but
- the generalized Rao bound for nets shows that 32^mÂ â‰¥ 109 867372 306873 634096 651188 437795 010157 566762 072796 965228 232709 312995 511352 868307 344802 883458 335094 340380 127588 137540 506774 936019 311162 806879 501480 991077 657824 > 32¹⁰⁵ [i]