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

Best Known (57, ∞, s)-Nets in Base 2

(57, ∞, 42)-Net over F₂ — Constructive and digital

Digital (57, m, 42)-net over F₂ for arbitrarily large m, using

net from sequence [i] based on digital (57, 41)-sequence over F₂, using
- t-expansion [i] based on digital (54, 41)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
    - an explicitly constructive algebraic function field [i]

(57, ∞, 66)-Net in Base 2 — Upper bound on s

There is no (57, m, 67)-net in base 2 for arbitrarily large m, because

m-reduction [i] would yield (57, 393, 67)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2³⁹³, 67, S₂, 6, 336), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 9 844827 660206 338998 063191 843261 482770 882003 335161 414642 666888 796811 219058 939207 054525 522909 380525 482349 380843 146099 818496 / 337 > 2³⁹³ [i]