MinT - Best Known (212−41, 212, s)-Nets in Base 2

Best Known (212−41, 212, s)-Nets in Base 2

(212−41, 212, 268)-Net over F₂ — Constructive and digital

Digital (171, 212, 268)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 24, 8)-net over F₂, using
  - net from sequence [i] based on digital (4, 7)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 4 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (147, 188, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 47, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(212−41, 212, 606)-Net over F₂ — Digital

Digital (171, 212, 606)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(212−41, 212, 12420)-Net in Base 2 — Upper bound on s

There is no (171, 212, 12421)-net in base 2, because

1 times m-reduction [i] would yield (171, 211, 12421)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3292 979615 884889 954711 887121 719671 205936 605966 624451 238719 266136 > 2²¹¹ [i]