MinT - Best Known (147−47, 147, s)-Nets in Base 2

Best Known (147−47, 147, s)-Nets in Base 2

(147−47, 147, 69)-Net over F₂ — Constructive and digital

Digital (100, 147, 69)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (0, 23, 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 (77, 124, 66)-net over F₂, using
  - trace code for nets [i] based on digital (15, 62, 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]

(147−47, 147, 72)-Net in Base 2 — Constructive

(100, 147, 72)-net in base 2, using

1 times m-reduction [i] based on (100, 148, 72)-net in base 2, using
- trace code for nets [i] based on (26, 74, 36)-net in base 4, using
  - net from sequence [i] based on (26, 35)-sequence in base 4, using
    - base expansion [i] based on digital (52, 35)-sequence over F₂, using
      - t-expansion [i] based on digital (51, 35)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 3 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(147−47, 147, 120)-Net over F₂ — Digital

Digital (100, 147, 120)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(147−47, 147, 734)-Net in Base 2 — Upper bound on s

There is no (100, 147, 735)-net in base 2, because

1 times m-reduction [i] would yield (100, 146, 735)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 90 178995 131478 971771 261778 486587 992746 501798 > 2¹⁴⁶ [i]