MinT - Best Known (207−89, 207, s)-Nets in Base 2

Best Known (207−89, 207, s)-Nets in Base 2

(207−89, 207, 62)-Net over F₂ — Constructive and digital

Digital (118, 207, 62)-net over F₂, using

1 times m-reduction [i] based on digital (118, 208, 62)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (19, 64, 20)-net over F₂, using
    - net from sequence [i] based on digital (19, 19)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 19 and N(F) ≥ 20, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 144, 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]

(207−89, 207, 81)-Net over F₂ — Digital

Digital (118, 207, 81)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(207−89, 207, 381)-Net in Base 2 — Upper bound on s

There is no (118, 207, 382)-net in base 2, because

1 times m-reduction [i] would yield (118, 206, 382)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 108 281135 681125 838540 035802 167193 363909 644684 259728 234437 582175 > 2²⁰⁶ [i]