MinT - Best Known (162−50, 162, s)-Nets in Base 2

Best Known (162−50, 162, s)-Nets in Base 2

(162−50, 162, 76)-Net over F₂ — Constructive and digital

Digital (112, 162, 76)-net over F₂, using

1 times m-reduction [i] based on digital (112, 163, 76)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 31, 10)-net over F₂, using
    - net from sequence [i] based on digital (6, 9)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 6 and N(F) ≥ 10, using
        an explicitly constructive algebraic function field [i]
    - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
  2. digital (81, 132, 66)-net over F₂, using
    - trace code for nets [i] based on digital (15, 66, 33)-net over F₄, using
      - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]

(162−50, 162, 86)-Net in Base 2 — Constructive

(112, 162, 86)-net in base 2, using

2 times m-reduction [i] based on (112, 164, 86)-net in base 2, using
- trace code for nets [i] based on (30, 82, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(162−50, 162, 141)-Net over F₂ — Digital

Digital (112, 162, 141)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−50, 162, 872)-Net in Base 2 — Upper bound on s

There is no (112, 162, 873)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5 989132 734572 257292 765352 516590 259813 134779 825036 > 2¹⁶² [i]