MinT - Best Known (239−46, 239, s)-Nets in Base 2

Best Known (239−46, 239, s)-Nets in Base 2

(239−46, 239, 271)-Net over F₂ — Constructive and digital

Digital (193, 239, 271)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 31, 11)-net over F₂, using
  - net from sequence [i] based on digital (8, 10)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 8 and N(F) ≥ 11, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (162, 208, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 52, 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]

(239−46, 239, 678)-Net over F₂ — Digital

Digital (193, 239, 678)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−46, 239, 12629)-Net in Base 2 — Upper bound on s

There is no (193, 239, 12630)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 884307 850829 703235 518814 448158 958144 762576 513604 447652 693202 793008 418236 > 2²³⁹ [i]