MinT - Best Known (69, 69+88, s)-Nets in Base 2

Best Known (69, 69+88, s)-Nets in Base 2

(69, 69+88, 48)-Net over F₂ — Constructive and digital

Digital (69, 157, 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+88, 49)-Net over F₂ — Digital

Digital (69, 157, 49)-net over F₂, using

t-expansion [i] based on digital (68, 157, 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+88, 147)-Net in Base 2 — Upper bound on s

There is no (69, 157, 148)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 210607 316318 595010 365802 096632 753052 492785 998448 > 2¹⁵⁷ [i]