MinT - Best Known (86−42, 86, s)-Nets in Base 2

Best Known (86−42, 86, s)-Nets in Base 2

(86−42, 86, 33)-Net over F₂ — Constructive and digital

Digital (44, 86, 33)-net over F₂, using

t-expansion [i] based on digital (39, 86, 33)-net over F₂, using
- net from sequence [i] based on digital (39, 32)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
    - an explicitly constructive algebraic function field [i]

(86−42, 86, 34)-Net over F₂ — Digital

Digital (44, 86, 34)-net over F₂, using

t-expansion [i] based on digital (43, 86, 34)-net over F₂, using
- net from sequence [i] based on digital (43, 33)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 43 and N(F) ≥ 34, using
    - a curve from the manYPoints database [i]

(86−42, 86, 102)-Net in Base 2 — Upper bound on s

There is no (44, 86, 103)-net in base 2, because

extracting embedded orthogonal array [i] would yield OA(2⁸⁶, 103, S₂, 42), but
- the linear programming bound shows that MÂ â‰¥ 158456 325028 528675 187087 900672 / 1705 > 2⁸⁶ [i]