MinT - Best Known (68, 111, s)-Nets in Base 2

Best Known (68, 111, s)-Nets in Base 2

(68, 111, 56)-Net over F₂ — Constructive and digital

Digital (68, 111, 56)-net over F₂, using

1 times m-reduction [i] based on digital (68, 112, 56)-net over F₂, using
- trace code for nets [i] based on digital (12, 56, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]

(68, 111, 62)-Net over F₂ — Digital

Digital (68, 111, 62)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 111, 297)-Net in Base 2 — Upper bound on s

There is no (68, 111, 298)-net in base 2, because

1 times m-reduction [i] would yield (68, 110, 298)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1310 731913 599681 178276 107722 706604 > 2¹¹⁰ [i]