MinT - Best Known (69, 258, s)-Nets in Base 2

Best Known (69, 258, s)-Nets in Base 2

(69, 258, 48)-Net over F₂ — Constructive and digital

Digital (69, 258, 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, 258, 49)-Net over F₂ — Digital

Digital (69, 258, 49)-net over F₂, using

t-expansion [i] based on digital (68, 258, 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, 258, 87)-Net in Base 2 — Upper bound on s

There is no (69, 258, 88)-net in base 2, because

3 times m-reduction [i] would yield (69, 255, 88)-net in base 2, but
- extracting embedded OOA [i] would yield OOA(2²⁵⁵, 88, S₂, 3, 186), but
  - the LP bound with quadratic polynomials shows that MÂ â‰¥ 32 884953 343397 799500 294159 742467 365830 328675 645041 920187 205953 858247 328817 741824 / 561 > 2²⁵⁵ [i]