MinT - Best Known (186−57, 186, s)-Nets in Base 2

Best Known (186−57, 186, s)-Nets in Base 2

(186−57, 186, 112)-Net over F₂ — Constructive and digital

Digital (129, 186, 112)-net over F₂, using

6 times m-reduction [i] based on digital (129, 192, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 96, 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]

(186−57, 186, 160)-Net over F₂ — Digital

Digital (129, 186, 160)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(186−57, 186, 1060)-Net in Base 2 — Upper bound on s

There is no (129, 186, 1061)-net in base 2, because

1 times m-reduction [i] would yield (129, 185, 1061)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 49 514793 551796 964505 846060 679690 368368 258782 972141 235572 > 2¹⁸⁵ [i]