MinT - Best Known (173−33, 173, s)-Nets in Base 2

Best Known (173−33, 173, s)-Nets in Base 2

(173−33, 173, 265)-Net over F₂ — Constructive and digital

Digital (140, 173, 265)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 17, 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 (123, 156, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 39, 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]

(173−33, 173, 542)-Net over F₂ — Digital

Digital (140, 173, 542)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(173−33, 173, 11687)-Net in Base 2 — Upper bound on s

There is no (140, 173, 11688)-net in base 2, because

1 times m-reduction [i] would yield (140, 172, 11688)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5986 432465 521623 867078 765005 332457 642663 778061 223622 > 2¹⁷² [i]