MinT - Best Known (252−101, 252, s)-Nets in Base 2

Best Known (252−101, 252, s)-Nets in Base 2

(252−101, 252, 76)-Net over F₂ — Constructive and digital

Digital (151, 252, 76)-net over F₂, using

3 times m-reduction [i] based on digital (151, 255, 76)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (45, 97, 34)-net over F₂, using
    - net from sequence [i] based on digital (45, 33)-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 1 place 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]
  2. digital (54, 158, 42)-net over F₂, using
    - net from sequence [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]

(252−101, 252, 112)-Net over F₂ — Digital

Digital (151, 252, 112)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−101, 252, 561)-Net in Base 2 — Upper bound on s

There is no (151, 252, 562)-net in base 2, because

1 times m-reduction [i] would yield (151, 251, 562)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3781 821205 171744 500580 746100 904706 813093 231271 977308 891560 595736 271753 124808 > 2²⁵¹ [i]