MinT - Best Known (68, 68+45, s)-Nets in Base 2

Best Known (68, 68+45, s)-Nets in Base 2

(68, 68+45, 54)-Net over F₂ — Constructive and digital

Digital (68, 113, 54)-net over F₂, using

3 times m-reduction [i] based on digital (68, 116, 54)-net over F₂, using
- trace code for nets [i] based on digital (10, 58, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]

(68, 68+45, 59)-Net over F₂ — Digital

Digital (68, 113, 59)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+45, 277)-Net in Base 2 — Upper bound on s

There is no (68, 113, 278)-net in base 2, because

1 times m-reduction [i] would yield (68, 112, 278)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5360 154458 226102 706938 865549 087128 > 2¹¹² [i]