MinT - Best Known (241−47, 241, s)-Nets in Base 2

Best Known (241−47, 241, s)-Nets in Base 2

(241−47, 241, 270)-Net over F₂ — Constructive and digital

Digital (194, 241, 270)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (6, 29, 10)-net over F₂, using
  - net from sequence [i] based on digital (6, 9)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 6 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (165, 212, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 53, 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]

(241−47, 241, 655)-Net over F₂ — Digital

Digital (194, 241, 655)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(241−47, 241, 13016)-Net in Base 2 — Upper bound on s

There is no (194, 241, 13017)-net in base 2, because

1 times m-reduction [i] would yield (194, 240, 13017)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 767184 189364 077345 496738 980850 720664 289740 579463 318219 615280 914034 996480 > 2²⁴⁰ [i]