MinT - Best Known (172−66, 172, s)-Nets in Base 2

Best Known (172−66, 172, s)-Nets in Base 2

(172−66, 172, 66)-Net over F₂ — Constructive and digital

Digital (106, 172, 66)-net over F₂, using

10 times m-reduction [i] based on digital (106, 182, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 91, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]

(172−66, 172, 89)-Net over F₂ — Digital

Digital (106, 172, 89)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−66, 172, 441)-Net in Base 2 — Upper bound on s

There is no (106, 172, 442)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6378 921740 014791 651611 463645 157939 479194 522759 009805 > 2¹⁷² [i]