MinT - Best Known (128, 128+67, s)-Nets in Base 2

Best Known (128, 128+67, s)-Nets in Base 2

(128, 128+67, 75)-Net over F₂ — Constructive and digital

Digital (128, 195, 75)-net over F₂, using

3 times m-reduction [i] based on digital (128, 198, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 74, 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, 124, 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]

(128, 128+67, 86)-Net in Base 2 — Constructive

(128, 195, 86)-net in base 2, using

1 times m-reduction [i] based on (128, 196, 86)-net in base 2, using
- trace code for nets [i] based on (30, 98, 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]

(128, 128+67, 128)-Net over F₂ — Digital

Digital (128, 195, 128)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+67, 726)-Net in Base 2 — Upper bound on s

There is no (128, 195, 727)-net in base 2, because

1 times m-reduction [i] would yield (128, 194, 727)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 25201 026502 310246 789547 328610 338023 038188 034948 146471 731280 > 2¹⁹⁴ [i]