MinT - Best Known (127, 127+65, s)-Nets in Base 2

Best Known (127, 127+65, s)-Nets in Base 2

(127, 127+65, 75)-Net over F₂ — Constructive and digital

Digital (127, 192, 75)-net over F₂, using

3 times m-reduction [i] based on digital (127, 195, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 73, 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, 122, 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]

(127, 127+65, 86)-Net in Base 2 — Constructive

(127, 192, 86)-net in base 2, using

2 times m-reduction [i] based on (127, 194, 86)-net in base 2, using
- trace code for nets [i] based on (30, 97, 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]

(127, 127+65, 130)-Net over F₂ — Digital

Digital (127, 192, 130)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+65, 754)-Net in Base 2 — Upper bound on s

There is no (127, 192, 755)-net in base 2, because

1 times m-reduction [i] would yield (127, 191, 755)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3165 548142 528078 585727 432629 582477 212097 282878 270058 511789 > 2¹⁹¹ [i]