MinT - Best Known (256−212, 256, s)-Nets in Base 2

Best Known (256−212, 256, s)-Nets in Base 2

(256−212, 256, 33)-Net over F₂ — Constructive and digital

Digital (44, 256, 33)-net over F₂, using

t-expansion [i] based on digital (39, 256, 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]

(256−212, 256, 34)-Net over F₂ — Digital

Digital (44, 256, 34)-net over F₂, using

t-expansion [i] based on digital (43, 256, 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]

(256−212, 256, 54)-Net in Base 2 — Upper bound on s

There is no (44, 256, 55)-net in base 2, because

43 times m-reduction [i] would yield (44, 213, 55)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²¹³, 55, S₂, 4, 169), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 263280 729171 392966 744795 069209 176080 797237 738501 372778 135777 443840 / 17 > 2²¹³ [i]