MinT - Best Known (52, 52+27, s)-Nets in Base 2

Best Known (52, 52+27, s)-Nets in Base 2

(52, 52+27, 56)-Net over F₂ — Constructive and digital

Digital (52, 79, 56)-net over F₂, using

1 times m-reduction [i] based on digital (52, 80, 56)-net over F₂, using
- trace code for nets [i] based on digital (12, 40, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]

(52, 52+27, 64)-Net over F₂ — Digital

Digital (52, 79, 64)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+27, 344)-Net in Base 2 — Upper bound on s

There is no (52, 79, 345)-net in base 2, because

1 times m-reduction [i] would yield (52, 78, 345)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 311366 547827 863639 374824 > 2⁷⁸ [i]