MinT - Best Known (152−49, 152, s)-Nets in Base 2

Best Known (152−49, 152, s)-Nets in Base 2

(152−49, 152, 69)-Net over F₂ — Constructive and digital

Digital (103, 152, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 24, 3)-net over F₂, using
  - net from sequence [i] based on digital (0, 2)-sequence over F₂, using
    - Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 0 and N(F) ≥ 3, using
      - the rational function field F₂(x) [i]
    - Niederreiter sequence [i]
    - Sobol sequence [i]
2. digital (79, 128, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 64, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]

(152−49, 152, 84)-Net in Base 2 — Constructive

(103, 152, 84)-net in base 2, using

trace code for nets [i] based on (27, 76, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(152−49, 152, 120)-Net over F₂ — Digital

Digital (103, 152, 120)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(152−49, 152, 733)-Net in Base 2 — Upper bound on s

There is no (103, 152, 734)-net in base 2, because

1 times m-reduction [i] would yield (103, 151, 734)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 2926 946993 914041 487322 067922 387935 013741 196951 > 2¹⁵¹ [i]