MinT - Best Known (128−99, 128, s)-Nets in Base 2

Best Known (128−99, 128, s)-Nets in Base 2

(128−99, 128, 21)-Net over F₂ — Constructive and digital

Digital (29, 128, 21)-net over F₂, using

t-expansion [i] based on digital (21, 128, 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]

(128−99, 128, 25)-Net over F₂ — Digital

Digital (29, 128, 25)-net over F₂, using

t-expansion [i] based on digital (28, 128, 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]

(128−99, 128, 40)-Net in Base 2 — Upper bound on s

There is no (29, 128, 41)-net in base 2, because

12 times m-reduction [i] would yield (29, 116, 41)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹¹⁶, 41, S₂, 3, 87), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 1 163074 496311 801388 790831 177745 301504 / 11 > 2¹¹⁶ [i]