MinT - Best Known (86−31, 86, s)-Nets in Base 2

Best Known (86−31, 86, s)-Nets in Base 2

(86−31, 86, 56)-Net over F₂ — Constructive and digital

Digital (55, 86, 56)-net over F₂, using

trace code for nets [i] based on digital (12, 43, 28)-net over F₄, using
- net from sequence [i] based on digital (12, 27)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
    - an explicitly constructive algebraic function field [i]

(86−31, 86, 60)-Net over F₂ — Digital

Digital (55, 86, 60)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(86−31, 86, 304)-Net in Base 2 — Upper bound on s

There is no (55, 86, 305)-net in base 2, because

1 times m-reduction [i] would yield (55, 85, 305)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 38 796780 144524 144435 437568 > 2⁸⁵ [i]