MinT - Best Known (219−87, 219, s)-Nets in Base 2

Best Known (219−87, 219, s)-Nets in Base 2

(219−87, 219, 69)-Net over F₂ — Constructive and digital

Digital (132, 219, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (19, 62, 20)-net over F₂, using
  - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (70, 157, 49)-net over F₂, using
  - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(219−87, 219, 101)-Net over F₂ — Digital

Digital (132, 219, 101)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−87, 219, 506)-Net in Base 2 — Upper bound on s

There is no (132, 219, 507)-net in base 2, because

1 times m-reduction [i] would yield (132, 218, 507)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 448981 791144 393974 770655 360633 747096 602252 130115 896890 300104 562610 > 2²¹⁸ [i]