MinT - Best Known (69, 69+37, s)-Nets in Base 2

Best Known (69, 69+37, s)-Nets in Base 2

(69, 69+37, 66)-Net over F₂ — Constructive and digital

Digital (69, 106, 66)-net over F₂, using

2 times m-reduction [i] based on digital (69, 108, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 54, 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]

(69, 69+37, 76)-Net over F₂ — Digital

Digital (69, 106, 76)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+37, 404)-Net in Base 2 — Upper bound on s

There is no (69, 106, 405)-net in base 2, because

1 times m-reduction [i] would yield (69, 105, 405)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 40 768000 033969 797511 211853 366898 > 2¹⁰⁵ [i]