MinT - Best Known (167, 167+39, s)-Nets in Base 2

Best Known (167, 167+39, s)-Nets in Base 2

(167, 167+39, 270)-Net over F₂ — Constructive and digital

Digital (167, 206, 270)-net over F₂, using

2¹ times duplication [i] based on digital (166, 205, 270)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 25, 10)-net over F₂, using
    - net from sequence [i] based on digital (6, 9)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 6 and N(F) ≥ 10, using
        an explicitly constructive algebraic function field [i]
    - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
  2. digital (141, 180, 260)-net over F₂, using
    - trace code for nets [i] based on digital (6, 45, 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]

(167, 167+39, 635)-Net over F₂ — Digital

Digital (167, 206, 635)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(167, 167+39, 14005)-Net in Base 2 — Upper bound on s

There is no (167, 206, 14006)-net in base 2, because

1 times m-reduction [i] would yield (167, 205, 14006)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 51 444436 057661 308928 074548 103454 140418 667292 134507 394458 897870 > 2²⁰⁵ [i]