MinT - Best Known (211−91, 211, s)-Nets in Base 2

Best Known (211−91, 211, s)-Nets in Base 2

(211−91, 211, 63)-Net over F₂ — Constructive and digital

Digital (120, 211, 63)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (21, 66, 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, 145, 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]

(211−91, 211, 81)-Net over F₂ — Digital

Digital (120, 211, 81)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−91, 211, 384)-Net in Base 2 — Upper bound on s

There is no (120, 211, 385)-net in base 2, because

1 times m-reduction [i] would yield (120, 210, 385)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1656 458867 368235 008552 673433 922458 276171 772574 291690 202868 323136 > 2²¹⁰ [i]