MinT - Best Known (58, 58+∞, s)-Nets in Base 2

Best Known (58, 58+∞, s)-Nets in Base 2

(58, 58+∞, 42)-Net over F₂ — Constructive and digital

Digital (58, m, 42)-net over F₂ for arbitrarily large m, using

net from sequence [i] based on digital (58, 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]

(58, 58+∞, 67)-Net in Base 2 — Upper bound on s

There is no (58, m, 68)-net in base 2 for arbitrarily large m, because

m-reduction [i] would yield (58, 399, 68)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2³⁹⁹, 68, S₂, 6, 341), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 103 289995 123476 343586 236766 880120 474973 188231 713168 940513 226374 261625 904880 673647 785185 814131 205513 257436 126878 909899 735040 / 57 > 2³⁹⁹ [i]