MinT - Best Known (86, 127, s)-Nets in Base 2

Best Known (86, 127, s)-Nets in Base 2

(86, 127, 68)-Net over F₂ — Constructive and digital

Digital (86, 127, 68)-net over F₂, using

3 times m-reduction [i] based on digital (86, 130, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 65, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(86, 127, 104)-Net over F₂ — Digital

Digital (86, 127, 104)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 127, 625)-Net in Base 2 — Upper bound on s

There is no (86, 127, 626)-net in base 2, because

1 times m-reduction [i] would yield (86, 126, 626)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 86 597108 772451 287325 800783 471934 877576 > 2¹²⁶ [i]