MinT - Best Known (257−102, 257, s)-Nets in Base 2

Best Known (257−102, 257, s)-Nets in Base 2

(257−102, 257, 77)-Net over F₂ — Constructive and digital

Digital (155, 257, 77)-net over F₂, using

t-expansion [i] based on digital (154, 257, 77)-net over F₂, using
- 1 times m-reduction [i] based on digital (154, 258, 77)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (48, 100, 35)-net over F₂, using
      - net from sequence [i] based on digital (48, 34)-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 2 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]
    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]

(257−102, 257, 116)-Net over F₂ — Digital

Digital (155, 257, 116)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−102, 257, 580)-Net in Base 2 — Upper bound on s

There is no (155, 257, 581)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 237928 112140 412720 426095 490415 861193 036665 419671 237085 177824 372438 615449 746048 > 2²⁵⁷ [i]