MinT - Best Known (127−45, 127, s)-Nets in Base 2

Best Known (127−45, 127, s)-Nets in Base 2

(127−45, 127, 66)-Net over F₂ — Constructive and digital

Digital (82, 127, 66)-net over F₂, using

7 times m-reduction [i] based on digital (82, 134, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 67, 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]

(127−45, 127, 84)-Net over F₂ — Digital

Digital (82, 127, 84)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(127−45, 127, 448)-Net in Base 2 — Upper bound on s

There is no (82, 127, 449)-net in base 2, because

1 times m-reduction [i] would yield (82, 126, 449)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 88 545991 535725 243182 294957 072222 205344 > 2¹²⁶ [i]