MinT - Best Known (51−21, 51, s)-Nets in Base 2

Best Known (51−21, 51, s)-Nets in Base 2

(51−21, 51, 30)-Net over F₂ — Constructive and digital

Digital (30, 51, 30)-net over F₂, using

1 times m-reduction [i] based on digital (30, 52, 30)-net over F₂, using
- trace code for nets [i] based on digital (4, 26, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]

(51−21, 51, 32)-Net over F₂ — Digital

Digital (30, 51, 32)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−21, 51, 131)-Net in Base 2 — Upper bound on s

There is no (30, 51, 132)-net in base 2, because

1 times m-reduction [i] would yield (30, 50, 132)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1203 299754 367888 > 2⁵⁰ [i]