MinT - Best Known (86, 86+57, s)-Nets in Base 2

Best Known (86, 86+57, s)-Nets in Base 2

(86, 86+57, 60)-Net over F₂ — Constructive and digital

Digital (86, 143, 60)-net over F₂, using

3 times m-reduction [i] based on digital (86, 146, 60)-net over F₂, using
- trace code for nets [i] based on digital (13, 73, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(86, 86+57, 71)-Net over F₂ — Digital

Digital (86, 143, 71)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(86, 86+57, 340)-Net in Base 2 — Upper bound on s

There is no (86, 143, 341)-net in base 2, because

1 times m-reduction [i] would yield (86, 142, 341)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5 853238 070789 200260 017309 015279 888616 506040 > 2¹⁴² [i]