MinT - Best Known (164−45, 164, s)-Nets in Base 2

Best Known (164−45, 164, s)-Nets in Base 2

(164−45, 164, 135)-Net over F₂ — Constructive and digital

Digital (119, 164, 135)-net over F₂, using

1 times m-reduction [i] based on digital (119, 165, 135)-net over F₂, using
- trace code for nets [i] based on digital (9, 55, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]

(164−45, 164, 188)-Net over F₂ — Digital

Digital (119, 164, 188)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−45, 164, 1506)-Net in Base 2 — Upper bound on s

There is no (119, 164, 1507)-net in base 2, because

1 times m-reduction [i] would yield (119, 163, 1507)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 11 739948 422280 336468 087140 378004 494046 632334 439464 > 2¹⁶³ [i]