MinT - Best Known (51, 51+27, s)-Nets in Base 2

Best Known (51, 51+27, s)-Nets in Base 2

(51, 51+27, 56)-Net over F₂ — Constructive and digital

Digital (51, 78, 56)-net over F₂, using

trace code for nets [i] based on digital (12, 39, 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]

(51, 51+27, 62)-Net over F₂ — Digital

Digital (51, 78, 62)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+27, 325)-Net in Base 2 — Upper bound on s

There is no (51, 78, 326)-net in base 2, because

1 times m-reduction [i] would yield (51, 77, 326)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 154826 833383 083450 765112 > 2⁷⁷ [i]