MinT - Best Known (168−69, 168, s)-Nets in Base 2

Best Known (168−69, 168, s)-Nets in Base 2

(168−69, 168, 66)-Net over F₂ — Constructive and digital

Digital (99, 168, 66)-net over F₂, using

trace code for nets [i] based on digital (15, 84, 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]

(168−69, 168, 75)-Net over F₂ — Digital

Digital (99, 168, 75)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−69, 168, 359)-Net in Base 2 — Upper bound on s

There is no (99, 168, 360)-net in base 2, because

1 times m-reduction [i] would yield (99, 167, 360)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 193 141588 330951 779550 509326 380810 453999 830469 119963 > 2¹⁶⁷ [i]