MinT - Best Known (56, ∞, s)-Nets in Base 2

Best Known (56, ∞, s)-Nets in Base 2

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

Digital (56, m, 42)-net over F₂ for arbitrarily large m, using

net from sequence [i] based on digital (56, 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]

(56, ∞, 65)-Net in Base 2 — Upper bound on s

There is no (56, m, 66)-net in base 2 for arbitrarily large m, because

m-reduction [i] would yield (56, 387, 66)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2³⁸⁷, 66, S₂, 6, 331), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 39086 790146 823323 378580 807779 342464 894639 101356 301723 094604 707057 011755 997325 232926 522952 124845 836315 680174 966384 361472 / 83 > 2³⁸⁷ [i]