MinT - Best Known (189−64, 189, s)-Nets in Base 2

Best Known (189−64, 189, s)-Nets in Base 2

(189−64, 189, 75)-Net over F₂ — Constructive and digital

Digital (125, 189, 75)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 71, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (54, 118, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(189−64, 189, 86)-Net in Base 2 — Constructive

(125, 189, 86)-net in base 2, using

1 times m-reduction [i] based on (125, 190, 86)-net in base 2, using
- trace code for nets [i] based on (30, 95, 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]

(189−64, 189, 129)-Net over F₂ — Digital

Digital (125, 189, 129)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−64, 189, 720)-Net in Base 2 — Upper bound on s

There is no (125, 189, 721)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 788 652715 140923 474060 655835 302501 098189 667931 094742 877430 > 2¹⁸⁹ [i]