MinT - Best Known (257−103, 257, s)-Nets in Base 2

Best Known (257−103, 257, s)-Nets in Base 2

(257−103, 257, 77)-Net over F₂ — Constructive and digital

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]

(257−103, 257, 114)-Net over F₂ — Digital

Digital (154, 257, 114)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−103, 257, 571)-Net in Base 2 — Upper bound on s

There is no (154, 257, 572)-net in base 2, because

1 times m-reduction [i] would yield (154, 256, 572)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 116793 663231 071502 565887 057748 868881 714179 847457 075425 572503 215840 798746 824329 > 2²⁵⁶ [i]