MinT - Best Known (127−37, 127, s)-Nets in Base 2

Best Known (127−37, 127, s)-Nets in Base 2

(127−37, 127, 84)-Net over F₂ — Constructive and digital

Digital (90, 127, 84)-net over F₂, using

2¹ times duplication [i] based on digital (89, 126, 84)-net over F₂, using
- trace code for nets [i] based on digital (5, 42, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]

(127−37, 127, 132)-Net over F₂ — Digital

Digital (90, 127, 132)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−37, 127, 940)-Net in Base 2 — Upper bound on s

There is no (90, 127, 941)-net in base 2, because

1 times m-reduction [i] would yield (90, 126, 941)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 85 654164 313439 322877 252905 846140 601244 > 2¹²⁶ [i]