MinT - Best Known (172−67, 172, s)-Nets in Base 2

Best Known (172−67, 172, s)-Nets in Base 2

(172−67, 172, 66)-Net over F₂ — Constructive and digital

Digital (105, 172, 66)-net over F₂, using

8 times m-reduction [i] based on digital (105, 180, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 90, 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]

(172−67, 172, 87)-Net over F₂ — Digital

Digital (105, 172, 87)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−67, 172, 430)-Net in Base 2 — Upper bound on s

There is no (105, 172, 431)-net in base 2, because

1 times m-reduction [i] would yield (105, 171, 431)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3000 500255 570079 375167 837131 076345 284184 362768 773776 > 2¹⁷¹ [i]