MinT - Best Known (69, 69+140, s)-Nets in Base 2

Best Known (69, 69+140, s)-Nets in Base 2

(69, 69+140, 48)-Net over F₂ — Constructive and digital

Digital (69, 209, 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+140, 49)-Net over F₂ — Digital

Digital (69, 209, 49)-net over F₂, using

t-expansion [i] based on digital (68, 209, 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+140, 101)-Net in Base 2 — Upper bound on s

There is no (69, 209, 102)-net in base 2, because

12 times m-reduction [i] would yield (69, 197, 102)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2¹⁹⁷, 102, S₂, 2, 128), but
  - the (dual) Plotkin bound for OOAs shows that MÂ â‰¥ 12 855504 354071 922204 335696 738729 300820 177623 950262 342682 411008 / 43 > 2¹⁹⁷ [i]