MinT - Best Known (206−76, 206, s)-Nets in Base 2

Best Known (206−76, 206, s)-Nets in Base 2

(206−76, 206, 70)-Net over F₂ — Constructive and digital

Digital (130, 206, 70)-net over F₂, using

2 times m-reduction [i] based on digital (130, 208, 70)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 60, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (70, 148, 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]

(206−76, 206, 84)-Net in Base 2 — Constructive

(130, 206, 84)-net in base 2, using

trace code for nets [i] based on (27, 103, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(206−76, 206, 114)-Net over F₂ — Digital

Digital (130, 206, 114)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(206−76, 206, 589)-Net in Base 2 — Upper bound on s

There is no (130, 206, 590)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 106 602358 353807 053931 311625 320608 036547 074505 883935 873315 420200 > 2²⁰⁶ [i]