MinT - Best Known (54, 54+31, s)-Nets in Base 2

Best Known (54, 54+31, s)-Nets in Base 2

(54, 54+31, 54)-Net over F₂ — Constructive and digital

Digital (54, 85, 54)-net over F₂, using

3 times m-reduction [i] based on digital (54, 88, 54)-net over F₂, using
- trace code for nets [i] based on digital (10, 44, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]

(54, 54+31, 58)-Net over F₂ — Digital

Digital (54, 85, 58)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+31, 290)-Net in Base 2 — Upper bound on s

There is no (54, 85, 291)-net in base 2, because

1 times m-reduction [i] would yield (54, 84, 291)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 20 071560 213100 950659 458288 > 2⁸⁴ [i]