MinT - Best Known (250−221, 250, s)-Nets in Base 2

Best Known (250−221, 250, s)-Nets in Base 2

(250−221, 250, 21)-Net over F₂ — Constructive and digital

Digital (29, 250, 21)-net over F₂, using

t-expansion [i] based on digital (21, 250, 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]

(250−221, 250, 25)-Net over F₂ — Digital

Digital (29, 250, 25)-net over F₂, using

t-expansion [i] based on digital (28, 250, 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]

(250−221, 250, 37)-Net in Base 2 — Upper bound on s

There is no (29, 250, 38)-net in base 2, because

68 times m-reduction [i] would yield (29, 182, 38)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁸², 38, S₂, 5, 153), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 637 518145 000209 765077 072363 294528 620385 395474 828378 505216 / 77 > 2¹⁸² [i]