MinT - Best Known (251−58, 251, s)-Nets in Base 2

Best Known (251−58, 251, s)-Nets in Base 2

(251−58, 251, 205)-Net over F₂ — Constructive and digital

Digital (193, 251, 205)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 35, 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]
2. digital (158, 216, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 72, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]

(251−58, 251, 418)-Net over F₂ — Digital

Digital (193, 251, 418)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−58, 251, 4662)-Net in Base 2 — Upper bound on s

There is no (193, 251, 4663)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3629 843129 299173 980845 471180 889233 588124 521421 402710 878538 378798 580190 373392 > 2²⁵¹ [i]