MinT - Best Known (128, 128+57, s)-Nets in Base 2

Best Known (128, 128+57, s)-Nets in Base 2

(128, 128+57, 112)-Net over F₂ — Constructive and digital

Digital (128, 185, 112)-net over F₂, using

5 times m-reduction [i] based on digital (128, 190, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 95, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(128, 128+57, 157)-Net over F₂ — Digital

Digital (128, 185, 157)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(128, 128+57, 1033)-Net in Base 2 — Upper bound on s

There is no (128, 185, 1034)-net in base 2, because

1 times m-reduction [i] would yield (128, 184, 1034)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 24 707971 783759 562033 117098 242855 821864 803825 359188 116224 > 2¹⁸⁴ [i]