MinT - Best Known (168, 168+41, s)-Nets in Base 2

Best Known (168, 168+41, s)-Nets in Base 2

(168, 168+41, 265)-Net over F₂ — Constructive and digital

Digital (168, 209, 265)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (147, 188, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 47, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(168, 168+41, 573)-Net over F₂ — Digital

Digital (168, 209, 573)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(168, 168+41, 11191)-Net in Base 2 — Upper bound on s

There is no (168, 209, 11192)-net in base 2, because

1 times m-reduction [i] would yield (168, 208, 11192)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 411 970712 298925 173101 190499 873201 242591 235649 719702 026396 402997 > 2²⁰⁸ [i]