MinT - Best Known (247−47, 247, s)-Nets in Base 2

Best Known (247−47, 247, s)-Nets in Base 2

(247−47, 247, 274)-Net over F₂ — Constructive and digital

Digital (200, 247, 274)-net over F₂, using

2¹ times duplication [i] based on digital (199, 246, 274)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 34, 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 (165, 212, 260)-net over F₂, using
    - trace code for nets [i] based on digital (6, 53, 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]

(247−47, 247, 723)-Net over F₂ — Digital

Digital (200, 247, 723)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(247−47, 247, 15603)-Net in Base 2 — Upper bound on s

There is no (200, 247, 15604)-net in base 2, because

1 times m-reduction [i] would yield (200, 246, 15604)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 113 149290 413820 157825 860026 892431 942277 462859 537764 743392 830058 100955 594568 > 2²⁴⁶ [i]