MinT - Best Known (224−85, 224, s)-Nets in Base 2

Best Known (224−85, 224, s)-Nets in Base 2

(224−85, 224, 75)-Net over F₂ — Constructive and digital

Digital (139, 224, 75)-net over F₂, using

7 times m-reduction [i] based on digital (139, 231, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 85, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 146, 42)-net over F₂, using
    - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]

(224−85, 224, 84)-Net in Base 2 — Constructive

(139, 224, 84)-net in base 2, using

trace code for nets [i] based on (27, 112, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(224−85, 224, 114)-Net over F₂ — Digital

Digital (139, 224, 114)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(224−85, 224, 594)-Net in Base 2 — Upper bound on s

There is no (139, 224, 595)-net in base 2, because

1 times m-reduction [i] would yield (139, 223, 595)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 13 545926 269360 019215 774970 907800 420046 914975 942223 011216 009429 687220 > 2²²³ [i]