MinT - Best Known (172−135, 172, s)-Nets in Base 2

Best Known (172−135, 172, s)-Nets in Base 2

(172−135, 172, 24)-Net over F₂ — Constructive and digital

Digital (37, 172, 24)-net over F₂, using

t-expansion [i] based on digital (33, 172, 24)-net over F₂, using
- net from sequence [i] based on digital (33, 23)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 33 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(172−135, 172, 30)-Net over F₂ — Digital

Digital (37, 172, 30)-net over F₂, using

t-expansion [i] based on digital (36, 172, 30)-net over F₂, using
- net from sequence [i] based on digital (36, 29)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 36 and N(F) ≥ 30, using
    - Drinfeld modules of rank 1 [i]

(172−135, 172, 49)-Net in Base 2 — Upper bound on s

There is no (37, 172, 50)-net in base 2, because

28 times m-reduction [i] would yield (37, 144, 50)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁴⁴, 50, S₂, 3, 107), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 624 420865 558857 447963 000111 634154 122167 451648 / 27 > 2¹⁴⁴ [i]