MinT - Best Known (40, ∞, s)-Nets in Base 2

Best Known (40, ∞, s)-Nets in Base 2

(40, ∞, 33)-Net over F₂ — Constructive and digital

Digital (40, m, 33)-net over F₂ for arbitrarily large m, using

net from sequence [i] based on digital (40, 32)-sequence over F₂, using
- t-expansion [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]

(40, ∞, 48)-Net in Base 2 — Upper bound on s

There is no (40, m, 49)-net in base 2 for arbitrarily large m, because

m-reduction [i] would yield (40, 286, 49)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁸⁶, 49, S₂, 6, 246), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 39288 535676 373144 730275 197603 598846 367936 035912 990469 677796 654174 559600 263990 136667 111424 / 247 > 2²⁸⁶ [i]