MinT - Best Known (172, 172+51, s)-Nets in Base 2

Best Known (172, 172+51, s)-Nets in Base 2

(172, 172+51, 202)-Net over F₂ — Constructive and digital

Digital (172, 223, 202)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (3, 28, 7)-net over F₂, using
  - net from sequence [i] based on digital (3, 6)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 3 and N(F) ≥ 7, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (144, 195, 195)-net over F₂, using
  - trace code for nets [i] based on digital (14, 65, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]

(172, 172+51, 388)-Net over F₂ — Digital

Digital (172, 223, 388)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+51, 4757)-Net in Base 2 — Upper bound on s

There is no (172, 223, 4758)-net in base 2, because

1 times m-reduction [i] would yield (172, 222, 4758)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6 741791 761735 690581 787086 176264 811053 177303 021548 802226 775607 103152 > 2²²² [i]