MinT - Best Known (135−57, 135, s)-Nets in Base 2

Best Known (135−57, 135, s)-Nets in Base 2

(135−57, 135, 54)-Net over F₂ — Constructive and digital

Digital (78, 135, 54)-net over F₂, using

1 times m-reduction [i] based on digital (78, 136, 54)-net over F₂, using
- trace code for nets [i] based on digital (10, 68, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]

(135−57, 135, 59)-Net over F₂ — Digital

Digital (78, 135, 59)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−57, 135, 272)-Net in Base 2 — Upper bound on s

There is no (78, 135, 273)-net in base 2, because

1 times m-reduction [i] would yield (78, 134, 273)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 22809 460935 517291 525033 720278 928207 382976 > 2¹³⁴ [i]