MinT - Best Known (260−51, 260, s)-Nets in Base 2

Best Known (260−51, 260, s)-Nets in Base 2

(260−51, 260, 270)-Net over F₂ — Constructive and digital

Digital (209, 260, 270)-net over F₂, using

2¹ times duplication [i] based on digital (208, 259, 270)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 31, 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 (177, 228, 260)-net over F₂, using
    - trace code for nets [i] based on digital (6, 57, 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]

(260−51, 260, 685)-Net over F₂ — Digital

Digital (209, 260, 685)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(260−51, 260, 13338)-Net in Base 2 — Upper bound on s

There is no (209, 260, 13339)-net in base 2, because

1 times m-reduction [i] would yield (209, 259, 13339)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 927839 586381 011093 681018 181195 637436 364837 432211 781570 801371 612157 072947 149140 > 2²⁵⁹ [i]