MinT - Best Known (69, 69+111, s)-Nets in Base 2

Best Known (69, 69+111, s)-Nets in Base 2

(69, 69+111, 48)-Net over F₂ — Constructive and digital

Digital (69, 180, 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+111, 49)-Net over F₂ — Digital

Digital (69, 180, 49)-net over F₂, using

t-expansion [i] based on digital (68, 180, 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+111, 134)-Net in Base 2 — Upper bound on s

There is no (69, 180, 135)-net in base 2, because

1 times m-reduction [i] would yield (69, 179, 135)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 984731 728315 551125 801265 688041 743087 457889 530369 154352 > 2¹⁷⁹ [i]