MinT - Best Known (137, 137+109, s)-Nets in Base 2

Best Known (137, 137+109, s)-Nets in Base 2

(137, 137+109, 66)-Net over F₂ — Constructive and digital

Digital (137, 246, 66)-net over F₂, using

9 times m-reduction [i] based on digital (137, 255, 66)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 98, 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 (39, 157, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂ (see above)

(137, 137+109, 87)-Net over F₂ — Digital

Digital (137, 246, 87)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+109, 411)-Net in Base 2 — Upper bound on s

There is no (137, 246, 412)-net in base 2, because

1 times m-reduction [i] would yield (137, 245, 412)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 57 405811 970896 815858 634397 984805 239549 713763 835084 015253 296165 319824 068344 > 2²⁴⁵ [i]