MinT - Best Known (226−86, 226, s)-Nets in Base 2

Best Known (226−86, 226, s)-Nets in Base 2

(226−86, 226, 75)-Net over F₂ — Constructive and digital

Digital (140, 226, 75)-net over F₂, using

8 times m-reduction [i] based on digital (140, 234, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 86, 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, 148, 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]

(226−86, 226, 84)-Net in Base 2 — Constructive

(140, 226, 84)-net in base 2, using

trace code for nets [i] based on (27, 113, 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]

(226−86, 226, 115)-Net over F₂ — Digital

Digital (140, 226, 115)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−86, 226, 583)-Net in Base 2 — Upper bound on s

There is no (140, 226, 584)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 108 884571 738866 631206 146943 921238 242318 749898 568521 072632 294986 625560 > 2²²⁶ [i]