MinT - Best Known (172, 172+42, s)-Nets in Base 2

Best Known (172, 172+42, s)-Nets in Base 2

(172, 172+42, 265)-Net over F₂ — Constructive and digital

Digital (172, 214, 265)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 5)-net over F₂, using
  - net from sequence [i] based on digital (1, 4)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 1 and N(F) ≥ 5, 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]

(172, 172+42, 583)-Net over F₂ — Digital

Digital (172, 214, 583)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(172, 172+42, 10111)-Net in Base 2 — Upper bound on s

There is no (172, 214, 10112)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 26370 127847 103875 766234 004301 431919 556380 346577 623424 247596 839417 > 2²¹⁴ [i]