MinT - Best Known (57−19, 57, s)-Nets in Base 2

Best Known (57−19, 57, s)-Nets in Base 2

(57−19, 57, 44)-Net over F₂ — Constructive and digital

Digital (38, 57, 44)-net over F₂, using

1 times m-reduction [i] based on digital (38, 58, 44)-net over F₂, using
- trace code for nets [i] based on digital (9, 29, 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]

(57−19, 57, 55)-Net over F₂ — Digital

Digital (38, 57, 55)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−19, 57, 296)-Net in Base 2 — Upper bound on s

There is no (38, 57, 297)-net in base 2, because

1 times m-reduction [i] would yield (38, 56, 297)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 72358 359571 402378 > 2⁵⁶ [i]