MinT - Best Known (137−47, 137, s)-Nets in Base 2

Best Known (137−47, 137, s)-Nets in Base 2

(137−47, 137, 68)-Net over F₂ — Constructive and digital

Digital (90, 137, 68)-net over F₂, using

1 times m-reduction [i] based on digital (90, 138, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 69, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(137−47, 137, 96)-Net over F₂ — Digital

Digital (90, 137, 96)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−47, 137, 535)-Net in Base 2 — Upper bound on s

There is no (90, 137, 536)-net in base 2, because

1 times m-reduction [i] would yield (90, 136, 536)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 90552 340478 576014 138488 755829 055783 475487 > 2¹³⁶ [i]