MinT - Best Known (123, 123+89, s)-Nets in Base 2

Best Known (123, 123+89, s)-Nets in Base 2

(123, 123+89, 66)-Net over F₂ — Constructive and digital

Digital (123, 212, 66)-net over F₂, using

4 times m-reduction [i] based on digital (123, 216, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 108, 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]

(123, 123+89, 87)-Net over F₂ — Digital

Digital (123, 212, 87)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+89, 417)-Net in Base 2 — Upper bound on s

There is no (123, 212, 418)-net in base 2, because

1 times m-reduction [i] would yield (123, 211, 418)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3449 277104 285364 931400 270001 104569 627428 545695 892908 924420 481376 > 2²¹¹ [i]