MinT - Best Known (168−145, 168, s)-Nets in Base 2

Best Known (168−145, 168, s)-Nets in Base 2

(168−145, 168, 21)-Net over F₂ — Constructive and digital

Digital (23, 168, 21)-net over F₂, using

t-expansion [i] based on digital (21, 168, 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]

(168−145, 168, 22)-Net over F₂ — Digital

Digital (23, 168, 22)-net over F₂, using

net from sequence [i] based on digital (23, 21)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 23 and N(F) ≥ 22, using
  - Drinfeld modules of rank 1 [i]

(168−145, 168, 30)-Net in Base 2 — Upper bound on s

There is no (23, 168, 31)-net in base 2, because

20 times m-reduction [i] would yield (23, 148, 31)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁴⁸, 31, S₂, 5, 125), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 7849 862309 882779 345820 572831 972223 250105 106432 / 21 > 2¹⁴⁸ [i]