MinT - Best Known (55−25, 55, s)-Nets in Base 2

Best Known (55−25, 55, s)-Nets in Base 2

(55−25, 55, 25)-Net over F₂ — Constructive and digital

Digital (30, 55, 25)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (7, 19, 11)-net over F₂, using
  - linear code embedding found in a computer search [i]
2. digital (11, 36, 14)-net over F₂, using
  - net from sequence [i] based on digital (11, 13)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 11 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]

(55−25, 55, 27)-Net over F₂ — Digital

Digital (30, 55, 27)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−25, 55, 103)-Net in Base 2 — Upper bound on s

There is no (30, 55, 104)-net in base 2, because

1 times m-reduction [i] would yield (30, 54, 104)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 19440 453672 426996 > 2⁵⁴ [i]