MinT - Best Known (53−19, 53, s)-Nets in Base 2

Best Known (53−19, 53, s)-Nets in Base 2

(53−19, 53, 42)-Net over F₂ — Constructive and digital

Digital (34, 53, 42)-net over F₂, using

1 times m-reduction [i] based on digital (34, 54, 42)-net over F₂, using
- trace code for nets [i] based on digital (7, 27, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]

(53−19, 53, 44)-Net over F₂ — Digital

Digital (34, 53, 44)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(53−19, 53, 214)-Net in Base 2 — Upper bound on s

There is no (34, 53, 215)-net in base 2, because

1 times m-reduction [i] would yield (34, 52, 215)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 4530 369805 615744 > 2⁵² [i]