MinT - Best Known (231−209, 231, s)-Nets in Base 2

Best Known (231−209, 231, s)-Nets in Base 2

(231−209, 231, 21)-Net over F₂ — Constructive and digital

Digital (22, 231, 21)-net over F₂, using

t-expansion [i] based on digital (21, 231, 21)-net over F₂, using
- net from sequence [i] based on digital (21, 20)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
    - an explicitly constructive algebraic function field [i]

(231−209, 231, 29)-Net in Base 2 — Upper bound on s

There is no (22, 231, 30)-net in base 2, because

88 times m-reduction [i] would yield (22, 143, 30)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁴³, 30, S₂, 5, 121), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 758 225336 750041 186812 214421 270044 291203 334144 / 61 > 2¹⁴³ [i]