MinT - Best Known (155−50, 155, s)-Nets in Base 2

Best Known (155−50, 155, s)-Nets in Base 2

(155−50, 155, 69)-Net over F₂ — Constructive and digital

Digital (105, 155, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 25, 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 (80, 130, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 65, 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]

(155−50, 155, 84)-Net in Base 2 — Constructive

(105, 155, 84)-net in base 2, using

1 times m-reduction [i] based on (105, 156, 84)-net in base 2, using
- trace code for nets [i] based on (27, 78, 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]

(155−50, 155, 122)-Net over F₂ — Digital

Digital (105, 155, 122)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(155−50, 155, 712)-Net in Base 2 — Upper bound on s

There is no (105, 155, 713)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 47219 655940 333511 761571 384463 191053 537559 871960 > 2¹⁵⁵ [i]