MinT - Best Known (251−221, 251, s)-Nets in Base 2

Best Known (251−221, 251, s)-Nets in Base 2

(251−221, 251, 21)-Net over F₂ — Constructive and digital

Digital (30, 251, 21)-net over F₂, using

t-expansion [i] based on digital (21, 251, 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]

(251−221, 251, 25)-Net over F₂ — Digital

Digital (30, 251, 25)-net over F₂, using

t-expansion [i] based on digital (28, 251, 25)-net over F₂, using
- net from sequence [i] based on digital (28, 24)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 28 and N(F) ≥ 25, using
    - a curve from the manYPoints database [i]

(251−221, 251, 38)-Net in Base 2 — Upper bound on s

There is no (30, 251, 39)-net in base 2, because

64 times m-reduction [i] would yield (30, 187, 39)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁸⁷, 39, S₂, 5, 157), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 19615 942923 083377 386986 841947 523957 550319 860763 950107 852800 / 79 > 2¹⁸⁷ [i]