MinT - Best Known (69, 69+123, s)-Nets in Base 2

Best Known (69, 69+123, s)-Nets in Base 2

(69, 69+123, 48)-Net over F₂ — Constructive and digital

Digital (69, 192, 48)-net over F₂, using

net from sequence [i] based on digital (69, 47)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using
  - an explicitly constructive algebraic function field [i]

(69, 69+123, 49)-Net over F₂ — Digital

Digital (69, 192, 49)-net over F₂, using

t-expansion [i] based on digital (68, 192, 49)-net over F₂, using
- net from sequence [i] based on digital (68, 48)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 68 and N(F) ≥ 49, using
    - an algebraic function field by Auer [i]

(69, 69+123, 130)-Net in Base 2 — Upper bound on s

There is no (69, 192, 131)-net in base 2, because

1 times m-reduction [i] would yield (69, 191, 131)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3830 788399 415638 311183 662139 896404 439007 841317 195009 736704 > 2¹⁹¹ [i]