MinT - Best Known (165−52, 165, s)-Nets in Base 2

Best Known (165−52, 165, s)-Nets in Base 2

(165−52, 165, 75)-Net over F₂ — Constructive and digital

Digital (113, 165, 75)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (5, 31, 9)-net over F₂, using
  - net from sequence [i] based on digital (5, 8)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 5 and N(F) ≥ 9, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (82, 134, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 67, 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]

(165−52, 165, 86)-Net in Base 2 — Constructive

(113, 165, 86)-net in base 2, using

1 times m-reduction [i] based on (113, 166, 86)-net in base 2, using
- trace code for nets [i] based on (30, 83, 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]

(165−52, 165, 136)-Net over F₂ — Digital

Digital (113, 165, 136)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−52, 165, 820)-Net in Base 2 — Upper bound on s

There is no (113, 165, 821)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 47 278210 916402 567974 876026 355344 991949 782149 673048 > 2¹⁶⁵ [i]