MinT - Best Known (56−19, 56, s)-Nets in Base 2

Best Known (56−19, 56, s)-Nets in Base 2

(56−19, 56, 44)-Net over F₂ — Constructive and digital

Digital (37, 56, 44)-net over F₂, using

trace code for nets [i] based on digital (9, 28, 22)-net over F₄, using
- net from sequence [i] based on digital (9, 21)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
    - an explicitly constructive algebraic function field [i]

(56−19, 56, 52)-Net over F₂ — Digital

Digital (37, 56, 52)-net over F₂, using

trace code for nets [i] based on digital (9, 28, 26)-net over F₄, using
- net from sequence [i] based on digital (9, 25)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 26, using
    - Drinfeld modules of rank 1 [i]

(56−19, 56, 273)-Net in Base 2 — Upper bound on s

There is no (37, 56, 274)-net in base 2, because

1 times m-reduction [i] would yield (37, 55, 274)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 36113 948239 922402 > 2⁵⁵ [i]