MinT - Best Known (131, 131+81, s)-Nets in Base 2

Best Known (131, 131+81, s)-Nets in Base 2

(131, 131+81, 70)-Net over F₂ — Constructive and digital

Digital (131, 212, 70)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (21, 61, 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, 151, 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]

(131, 131+81, 108)-Net over F₂ — Digital

Digital (131, 212, 108)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+81, 553)-Net in Base 2 — Upper bound on s

There is no (131, 212, 554)-net in base 2, because

1 times m-reduction [i] would yield (131, 211, 554)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3384 546426 224401 003884 128559 367117 479728 708130 145731 326593 712217 > 2²¹¹ [i]