MinT - Best Known (219−93, 219, s)-Nets in Base 2

Best Known (219−93, 219, s)-Nets in Base 2

(219−93, 219, 66)-Net over F₂ — Constructive and digital

Digital (126, 219, 66)-net over F₂, using

3 times m-reduction [i] based on digital (126, 222, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 111, 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]

(219−93, 219, 87)-Net over F₂ — Digital

Digital (126, 219, 87)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−93, 219, 416)-Net in Base 2 — Upper bound on s

There is no (126, 219, 417)-net in base 2, because

1 times m-reduction [i] would yield (126, 218, 417)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 449856 006030 559923 341831 919887 376002 321077 531188 972495 802141 757248 > 2²¹⁸ [i]