MinT - Best Known (81−58, 81, s)-Nets in Base 2

Best Known (81−58, 81, s)-Nets in Base 2

(81−58, 81, 21)-Net over F₂ — Constructive and digital

Digital (23, 81, 21)-net over F₂, using

t-expansion [i] based on digital (21, 81, 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]

(81−58, 81, 22)-Net over F₂ — Digital

Digital (23, 81, 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]

(81−58, 81, 38)-Net in Base 2 — Upper bound on s

There is no (23, 81, 39)-net in base 2, because

9 times m-reduction [i] would yield (23, 72, 39)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2⁷², 39, S₂, 2, 49), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 37778 931862 957161 709568 / 5 > 2⁷² [i]