MinT - Best Known (187−55, 187, s)-Nets in Base 2

Best Known (187−55, 187, s)-Nets in Base 2

(187−55, 187, 112)-Net over F₂ — Constructive and digital

Digital (132, 187, 112)-net over F₂, using

11 times m-reduction [i] based on digital (132, 198, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 99, 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]

(187−55, 187, 177)-Net over F₂ — Digital

Digital (132, 187, 177)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−55, 187, 1255)-Net in Base 2 — Upper bound on s

There is no (132, 187, 1256)-net in base 2, because

1 times m-reduction [i] would yield (132, 186, 1256)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 99 615775 973188 721523 652178 982313 372101 585947 126561 455308 > 2¹⁸⁶ [i]