MinT - Best Known (68−29, 68, s)-Nets in Base 2

Best Known (68−29, 68, s)-Nets in Base 2

(68−29, 68, 34)-Net over F₂ — Constructive and digital

Digital (39, 68, 34)-net over F₂, using

trace code for nets [i] based on digital (5, 34, 17)-net over F₄, using
- net from sequence [i] based on digital (5, 16)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
    - an explicitly constructive algebraic function field [i]

(68−29, 68, 35)-Net over F₂ — Digital

Digital (39, 68, 35)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−29, 68, 147)-Net in Base 2 — Upper bound on s

There is no (39, 68, 148)-net in base 2, because

1 times m-reduction [i] would yield (39, 67, 148)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 156 108134 979284 758764 > 2⁶⁷ [i]