MinT - Best Known (123, 218, s)-Nets in Base 2

Best Known (123, 218, s)-Nets in Base 2

(123, 218, 63)-Net over F₂ — Constructive and digital

Digital (123, 218, 63)-net over F₂, using

1 times m-reduction [i] based on digital (123, 219, 63)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 69, 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, 150, 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]

(123, 218, 82)-Net over F₂ — Digital

Digital (123, 218, 82)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 218, 385)-Net in Base 2 — Upper bound on s

There is no (123, 218, 386)-net in base 2, because

1 times m-reduction [i] would yield (123, 217, 386)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 225233 649974 835077 564411 315280 064954 033625 132579 038445 800864 910912 > 2²¹⁷ [i]