MinT - Best Known (180−53, 180, s)-Nets in Base 2

Best Known (180−53, 180, s)-Nets in Base 2

(180−53, 180, 112)-Net over F₂ — Constructive and digital

Digital (127, 180, 112)-net over F₂, using

8 times m-reduction [i] based on digital (127, 188, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 94, 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]

(180−53, 180, 171)-Net over F₂ — Digital

Digital (127, 180, 171)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(180−53, 180, 1208)-Net in Base 2 — Upper bound on s

There is no (127, 180, 1209)-net in base 2, because

1 times m-reduction [i] would yield (127, 179, 1209)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 770193 967597 668268 388807 054737 022177 673594 492756 930700 > 2¹⁷⁹ [i]