MinT - Best Known (135, 135+51, s)-Nets in Base 2

Best Known (135, 135+51, s)-Nets in Base 2

(135, 135+51, 144)-Net over F₂ — Constructive and digital

Digital (135, 186, 144)-net over F₂, using

trace code for nets [i] based on digital (11, 62, 48)-net over F₈, using
- net from sequence [i] based on digital (11, 47)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
    - a function field by SÃ©mirat [i]

(135, 135+51, 208)-Net over F₂ — Digital

Digital (135, 186, 208)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+51, 1682)-Net in Base 2 — Upper bound on s

There is no (135, 186, 1683)-net in base 2, because

1 times m-reduction [i] would yield (135, 185, 1683)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 49 578820 684145 364618 591218 783464 175997 473115 038565 321664 > 2¹⁸⁵ [i]