MinT - Best Known (191, 191+45, s)-Nets in Base 2

Best Known (191, 191+45, s)-Nets in Base 2

(191, 191+45, 272)-Net over F₂ — Constructive and digital

Digital (191, 236, 272)-net over F₂, using

2¹ times duplication [i] based on digital (190, 235, 272)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 31, 12)-net over F₂, using
    - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
        an explicitly constructive algebraic function field [i]
    - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
  2. digital (159, 204, 260)-net over F₂, using
    - trace code for nets [i] based on digital (6, 51, 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]

(191, 191+45, 692)-Net over F₂ — Digital

Digital (191, 236, 692)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(191, 191+45, 14840)-Net in Base 2 — Upper bound on s

There is no (191, 236, 14841)-net in base 2, because

1 times m-reduction [i] would yield (191, 235, 14841)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 55246 214185 256403 144985 603509 547044 729077 503964 439541 311055 381398 290228 > 2²³⁵ [i]