MinT - Best Known (222−86, 222, s)-Nets in Base 2

Best Known (222−86, 222, s)-Nets in Base 2

(222−86, 222, 75)-Net over F₂ — Constructive and digital

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]

(222−86, 222, 108)-Net over F₂ — Digital

Digital (136, 222, 108)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−86, 222, 543)-Net in Base 2 — Upper bound on s

There is no (136, 222, 544)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6 863054 284835 862240 155672 676906 448558 525807 530281 136330 584971 383632 > 2²²² [i]