MinT - Best Known (197−67, 197, s)-Nets in Base 2

Best Known (197−67, 197, s)-Nets in Base 2

(197−67, 197, 76)-Net over F₂ — Constructive and digital

Digital (130, 197, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (43, 76, 34)-net over F₂, using
  - trace code for nets [i] based on digital (5, 38, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
2. digital (54, 121, 42)-net over F₂, using
  - net from sequence [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]

(197−67, 197, 86)-Net in Base 2 — Constructive

(130, 197, 86)-net in base 2, using

3 times m-reduction [i] based on (130, 200, 86)-net in base 2, using
- trace code for nets [i] based on (30, 100, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(197−67, 197, 132)-Net over F₂ — Digital

Digital (130, 197, 132)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(197−67, 197, 760)-Net in Base 2 — Upper bound on s

There is no (130, 197, 761)-net in base 2, because

1 times m-reduction [i] would yield (130, 196, 761)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 104241 235113 387301 909875 200099 253957 634009 245680 387577 205914 > 2¹⁹⁶ [i]