MinT - Best Known (136, 217, s)-Nets in Base 2

Best Known (136, 217, s)-Nets in Base 2

(136, 217, 75)-Net over F₂ — Constructive and digital

Digital (136, 217, 75)-net over F₂, using

5 times m-reduction [i] based on digital (136, 222, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 82, 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, 140, 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]

(136, 217, 84)-Net in Base 2 — Constructive

(136, 217, 84)-net in base 2, using

1 times m-reduction [i] based on (136, 218, 84)-net in base 2, using
- trace code for nets [i] based on (27, 109, 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]

(136, 217, 116)-Net over F₂ — Digital

Digital (136, 217, 116)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(136, 217, 608)-Net in Base 2 — Upper bound on s

There is no (136, 217, 609)-net in base 2, because

1 times m-reduction [i] would yield (136, 216, 609)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 107656 437238 572290 575289 052719 137818 466677 598805 493133 738304 446440 > 2²¹⁶ [i]