MinT - Best Known (223−42, 223, s)-Nets in Base 2

Best Known (223−42, 223, s)-Nets in Base 2

(223−42, 223, 272)-Net over F₂ — Constructive and digital

Digital (181, 223, 272)-net over F₂, using

2¹ times duplication [i] based on digital (180, 222, 272)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 30, 12)-net over F₂, using
    - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
        an explicitly constructive algebraic function field [i]
    - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
  2. digital (150, 192, 260)-net over F₂, using
    - trace code for nets [i] based on digital (6, 48, 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]

(223−42, 223, 688)-Net over F₂ — Digital

Digital (181, 223, 688)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−42, 223, 13619)-Net in Base 2 — Upper bound on s

There is no (181, 223, 13620)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 13 492376 890600 093804 480694 943711 863806 656777 791122 251854 288515 258672 > 2²²³ [i]