MinT - Best Known (131, 131+67, s)-Nets in Base 2

Best Known (131, 131+67, s)-Nets in Base 2

(131, 131+67, 76)-Net over F₂ — Constructive and digital

Digital (131, 198, 76)-net over F₂, using

2¹ times duplication [i] based on 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]

(131, 131+67, 86)-Net in Base 2 — Constructive

(131, 198, 86)-net in base 2, using

4 times m-reduction [i] based on (131, 202, 86)-net in base 2, using
- trace code for nets [i] based on (30, 101, 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]

(131, 131+67, 134)-Net over F₂ — Digital

Digital (131, 198, 134)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(131, 131+67, 777)-Net in Base 2 — Upper bound on s

There is no (131, 198, 778)-net in base 2, because

1 times m-reduction [i] would yield (131, 197, 778)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 207324 004627 039510 193993 205250 982777 148675 844817 224050 432408 > 2¹⁹⁷ [i]