MinT - Best Known (199, 199+47, s)-Nets in Base 2

Best Known (199, 199+47, s)-Nets in Base 2

(199, 199+47, 274)-Net over F₂ — Constructive and digital

Digital (199, 246, 274)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (11, 34, 14)-net over F₂, using
  - net from sequence [i] based on digital (11, 13)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 11 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [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]

(199, 199+47, 711)-Net over F₂ — Digital

Digital (199, 246, 711)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(199, 199+47, 15139)-Net in Base 2 — Upper bound on s

There is no (199, 246, 15140)-net in base 2, because

1 times m-reduction [i] would yield (199, 245, 15140)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 56 595842 549789 378475 572245 228896 452954 265857 543955 014925 671040 403804 185544 > 2²⁴⁵ [i]