MinT - Best Known (55−19, 55, s)-Nets in Base 2

Best Known (55−19, 55, s)-Nets in Base 2

(55−19, 55, 42)-Net over F₂ — Constructive and digital

Digital (36, 55, 42)-net over F₂, using

3 times m-reduction [i] based on digital (36, 58, 42)-net over F₂, using
- trace code for nets [i] based on digital (7, 29, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]

(55−19, 55, 49)-Net over F₂ — Digital

Digital (36, 55, 49)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−19, 55, 252)-Net in Base 2 — Upper bound on s

There is no (36, 55, 253)-net in base 2, because

1 times m-reduction [i] would yield (36, 54, 253)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 18201 391112 031678 > 2⁵⁴ [i]