MinT - Best Known (217−42, 217, s)-Nets in Base 2

Best Known (217−42, 217, s)-Nets in Base 2

(217−42, 217, 268)-Net over F₂ — Constructive and digital

Digital (175, 217, 268)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 25, 8)-net over F₂, using
  - net from sequence [i] based on digital (4, 7)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 4 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (150, 192, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 48, 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]

(217−42, 217, 616)-Net over F₂ — Digital

Digital (175, 217, 616)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−42, 217, 11166)-Net in Base 2 — Upper bound on s

There is no (175, 217, 11167)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 210656 073819 312096 581874 780675 763295 147260 480041 538246 808663 339876 > 2²¹⁷ [i]