MinT - Best Known (68, 68+39, s)-Nets in Base 2

Best Known (68, 68+39, s)-Nets in Base 2

(68, 68+39, 60)-Net over F₂ — Constructive and digital

Digital (68, 107, 60)-net over F₂, using

3 times m-reduction [i] based on digital (68, 110, 60)-net over F₂, using
- trace code for nets [i] based on digital (13, 55, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(68, 68+39, 69)-Net over F₂ — Digital

Digital (68, 107, 69)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+39, 351)-Net in Base 2 — Upper bound on s

There is no (68, 107, 352)-net in base 2, because

1 times m-reduction [i] would yield (68, 106, 352)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 81 497073 562541 981188 126533 179159 > 2¹⁰⁶ [i]