MinT - Best Known (127, 127+45, s)-Nets in Base 2

Best Known (127, 127+45, s)-Nets in Base 2

(127, 127+45, 144)-Net over F₂ — Constructive and digital

Digital (127, 172, 144)-net over F₂, using

2 times m-reduction [i] based on digital (127, 174, 144)-net over F₂, using
- trace code for nets [i] based on digital (11, 58, 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]

(127, 127+45, 220)-Net over F₂ — Digital

Digital (127, 172, 220)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(127, 127+45, 1947)-Net in Base 2 — Upper bound on s

There is no (127, 172, 1948)-net in base 2, because

1 times m-reduction [i] would yield (127, 171, 1948)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2999 967895 152690 576881 149226 894111 459593 695206 593818 > 2¹⁷¹ [i]