MinT - Best Known (137, 137+83, s)-Nets in Base 2

Best Known (137, 137+83, s)-Nets in Base 2

(137, 137+83, 75)-Net over F₂ — Constructive and digital

Digital (137, 220, 75)-net over F₂, using

5 times m-reduction [i] based on digital (137, 225, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 83, 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, 142, 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]

(137, 137+83, 84)-Net in Base 2 — Constructive

(137, 220, 84)-net in base 2, using

trace code for nets [i] based on (27, 110, 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]

(137, 137+83, 114)-Net over F₂ — Digital

Digital (137, 220, 114)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+83, 595)-Net in Base 2 — Upper bound on s

There is no (137, 220, 596)-net in base 2, because

1 times m-reduction [i] would yield (137, 219, 596)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 848235 690014 989694 874435 171060 539986 344061 754180 563935 208299 736500 > 2²¹⁹ [i]