MinT - Best Known (219−86, 219, s)-Nets in Base 2

Best Known (219−86, 219, s)-Nets in Base 2

(219−86, 219, 69)-Net over F₂ — Constructive and digital

Digital (133, 219, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (39, 82, 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 (51, 137, 36)-net over F₂, using
  - net from sequence [i] based on digital (51, 35)-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 3 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]

(219−86, 219, 104)-Net over F₂ — Digital

Digital (133, 219, 104)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−86, 219, 515)-Net in Base 2 — Upper bound on s

There is no (133, 219, 516)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 885832 843308 488167 097504 475110 046116 245945 193357 250827 591371 242048 > 2²¹⁹ [i]