MinT - Best Known (54−15, 54, s)-Nets in Base 2

Best Known (54−15, 54, s)-Nets in Base 2

(54−15, 54, 72)-Net over F₂ — Constructive and digital

Digital (39, 54, 72)-net over F₂, using

trace code for nets [i] based on digital (3, 18, 24)-net over F₈, using
- net from sequence [i] based on digital (3, 23)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
    - the Klein quartic over F₈ [i]

(54−15, 54, 82)-Net over F₂ — Digital

Digital (39, 54, 82)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−15, 54, 632)-Net in Base 2 — Upper bound on s

There is no (39, 54, 633)-net in base 2, because

1 times m-reduction [i] would yield (39, 53, 633)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 9010 005759 115264 > 2⁵³ [i]