MinT - Best Known (134, 134+76, s)-Nets in Base 2

Best Known (134, 134+76, s)-Nets in Base 2

(134, 134+76, 75)-Net over F₂ — Constructive and digital

Digital (134, 210, 75)-net over F₂, using

6 times m-reduction [i] based on digital (134, 216, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 80, 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, 136, 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]

(134, 134+76, 84)-Net in Base 2 — Constructive

(134, 210, 84)-net in base 2, using

4 times m-reduction [i] based on (134, 214, 84)-net in base 2, using
- trace code for nets [i] based on (27, 107, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [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]

(134, 134+76, 121)-Net over F₂ — Digital

Digital (134, 210, 121)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(134, 134+76, 637)-Net in Base 2 — Upper bound on s

There is no (134, 210, 638)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1648 019408 104815 771501 970275 306181 814438 484692 078667 055236 777348 > 2²¹⁰ [i]