MinT - Best Known (255−112, 255, s)-Nets in Base 2

Best Known (255−112, 255, s)-Nets in Base 2

(255−112, 255, 68)-Net over F₂ — Constructive and digital

Digital (143, 255, 68)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 95, 33)-net over F₂, using
  - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (48, 160, 35)-net over F₂, using
  - net from sequence [i] based on digital (48, 34)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(255−112, 255, 92)-Net over F₂ — Digital

Digital (143, 255, 92)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−112, 255, 432)-Net in Base 2 — Upper bound on s

There is no (143, 255, 433)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 64319 838873 631844 977435 554690 000843 010954 380063 586616 901705 656681 825591 673776 > 2²⁵⁵ [i]