MinT - Best Known (257−99, 257, s)-Nets in Base 2

Best Known (257−99, 257, s)-Nets in Base 2

(257−99, 257, 84)-Net over F₂ — Constructive and digital

Digital (158, 257, 84)-net over F₂, using

1 times m-reduction [i] based on digital (158, 258, 84)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (54, 104, 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]
  2. digital (54, 154, 42)-net over F₂, using
    - net from sequence [i] based on digital (54, 41)-sequence over F₂ (see above)

(257−99, 257, 125)-Net over F₂ — Digital

Digital (158, 257, 125)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(257−99, 257, 644)-Net in Base 2 — Upper bound on s

There is no (158, 257, 645)-net in base 2, because

1 times m-reduction [i] would yield (158, 256, 645)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 117466 060290 398584 088039 468113 289537 095459 103339 491303 522767 623268 417976 983984 > 2²⁵⁶ [i]