MinT - Best Known (31, 31+165, s)-Nets in Base 2

Best Known (31, 31+165, s)-Nets in Base 2

(31, 31+165, 21)-Net over F₂ — Constructive and digital

Digital (31, 196, 21)-net over F₂, using

t-expansion [i] based on digital (21, 196, 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]

(31, 31+165, 27)-Net over F₂ — Digital

Digital (31, 196, 27)-net over F₂, using

net from sequence [i] based on digital (31, 26)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 31 and N(F) ≥ 27, using
  - Drinfeld modules of rank 1 [i]

(31, 31+165, 39)-Net in Base 2 — Upper bound on s

There is no (31, 196, 40)-net in base 2, because

4 times m-reduction [i] would yield (31, 192, 40)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁹², 40, S₂, 5, 161), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 200867 255532 373784 442745 261542 645325 315275 374222 849104 412672 / 27 > 2¹⁹² [i]