MinT - Best Known (135, 135+53, s)-Nets in Base 2

Best Known (135, 135+53, s)-Nets in Base 2

(135, 135+53, 135)-Net over F₂ — Constructive and digital

Digital (135, 188, 135)-net over F₂, using

1 times m-reduction [i] based on digital (135, 189, 135)-net over F₂, using
- trace code for nets [i] based on digital (9, 63, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]

(135, 135+53, 196)-Net over F₂ — Digital

Digital (135, 188, 196)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(135, 135+53, 1505)-Net in Base 2 — Upper bound on s

There is no (135, 188, 1506)-net in base 2, because

1 times m-reduction [i] would yield (135, 187, 1506)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 199 389433 019786 916901 410326 424254 875645 450563 870922 781104 > 2¹⁸⁷ [i]