MinT - Best Known (178−69, 178, s)-Nets in Base 2

Best Known (178−69, 178, s)-Nets in Base 2

(178−69, 178, 66)-Net over F₂ — Constructive and digital

Digital (109, 178, 66)-net over F₂, using

10 times m-reduction [i] based on digital (109, 188, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 94, 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]

(178−69, 178, 90)-Net over F₂ — Digital

Digital (109, 178, 90)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−69, 178, 451)-Net in Base 2 — Upper bound on s

There is no (109, 178, 452)-net in base 2, because

1 times m-reduction [i] would yield (109, 177, 452)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 201103 735153 624162 394248 395837 395114 856180 088372 337070 > 2¹⁷⁷ [i]