MinT - Best Known (138, 138+87, s)-Nets in Base 2

Best Known (138, 138+87, s)-Nets in Base 2

(138, 138+87, 75)-Net over F₂ — Constructive and digital

Digital (138, 225, 75)-net over F₂, using

3 times m-reduction [i] based on digital (138, 228, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 84, 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, 144, 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]

(138, 138+87, 110)-Net over F₂ — Digital

Digital (138, 225, 110)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+87, 563)-Net in Base 2 — Upper bound on s

There is no (138, 225, 564)-net in base 2, because

1 times m-reduction [i] would yield (138, 224, 564)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27 955353 141056 418667 359022 664727 940846 790188 092990 504906 523811 383286 > 2²²⁴ [i]