MinT - Best Known (257−50, 257, s)-Nets in Base 2

Best Known (257−50, 257, s)-Nets in Base 2

(257−50, 257, 271)-Net over F₂ — Constructive and digital

Digital (207, 257, 271)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 33, 11)-net over F₂, using
  - net from sequence [i] based on digital (8, 10)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 8 and N(F) ≥ 11, using
      - an explicitly constructive algebraic function field [i]
2. digital (174, 224, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 56, 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]

(257−50, 257, 696)-Net over F₂ — Digital

Digital (207, 257, 696)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−50, 257, 12616)-Net in Base 2 — Upper bound on s

There is no (207, 257, 12617)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 231743 574349 421461 302727 301459 722029 094868 403101 170020 878939 347349 212726 795176 > 2²⁵⁷ [i]