MinT - Best Known (256−49, 256, s)-Nets in Base 2

Best Known (256−49, 256, s)-Nets in Base 2

(256−49, 256, 274)-Net over F₂ — Constructive and digital

Digital (207, 256, 274)-net over F₂, using

2¹ times duplication [i] based on digital (206, 255, 274)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 35, 14)-net over F₂, using
    - net from sequence [i] based on digital (11, 13)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 11 and N(F) ≥ 14, using
        an explicitly constructive algebraic function field [i]
  2. digital (171, 220, 260)-net over F₂, using
    - trace code for nets [i] based on digital (6, 55, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(256−49, 256, 731)-Net over F₂ — Digital

Digital (207, 256, 731)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(256−49, 256, 15445)-Net in Base 2 — Upper bound on s

There is no (207, 256, 15446)-net in base 2, because

1 times m-reduction [i] would yield (207, 255, 15446)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 57913 819345 297495 429869 618794 110867 403990 544103 414390 516256 172305 492183 832376 > 2²⁵⁵ [i]