MinT - Best Known (219−91, 219, s)-Nets in Base 2

Best Known (219−91, 219, s)-Nets in Base 2

(219−91, 219, 66)-Net over F₂ — Constructive and digital

Digital (128, 219, 66)-net over F₂, using

7 times m-reduction [i] based on digital (128, 226, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 113, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]

(219−91, 219, 91)-Net over F₂ — Digital

Digital (128, 219, 91)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−91, 219, 443)-Net in Base 2 — Upper bound on s

There is no (128, 219, 444)-net in base 2, because

1 times m-reduction [i] would yield (128, 218, 444)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 453014 646107 609837 318917 144811 502361 159649 151419 114603 038514 644068 > 2²¹⁸ [i]