MinT - Best Known (172−68, 172, s)-Nets in Base 2

Best Known (172−68, 172, s)-Nets in Base 2

(172−68, 172, 66)-Net over F₂ — Constructive and digital

Digital (104, 172, 66)-net over F₂, using

6 times m-reduction [i] based on digital (104, 178, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 89, 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]

(172−68, 172, 84)-Net over F₂ — Digital

Digital (104, 172, 84)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−68, 172, 403)-Net in Base 2 — Upper bound on s

There is no (104, 172, 404)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6409 445578 352306 870453 715504 205732 927198 610687 843892 > 2¹⁷² [i]