MinT - Best Known (222−97, 222, s)-Nets in Base 2

Best Known (222−97, 222, s)-Nets in Base 2

(222−97, 222, 63)-Net over F₂ — Constructive and digital

Digital (125, 222, 63)-net over F₂, using

3 times m-reduction [i] based on digital (125, 225, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 71, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 154, 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]

(222−97, 222, 82)-Net over F₂ — Digital

Digital (125, 222, 82)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(222−97, 222, 389)-Net in Base 2 — Upper bound on s

There is no (125, 222, 390)-net in base 2, because

1 times m-reduction [i] would yield (125, 221, 390)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 733122 961999 397173 919402 667478 444632 042687 060149 630752 480271 256568 > 2²²¹ [i]