MinT - Best Known (248−49, 248, s)-Nets in Base 2

Best Known (248−49, 248, s)-Nets in Base 2

(248−49, 248, 268)-Net over F₂ — Constructive and digital

Digital (199, 248, 268)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 28, 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 (171, 220, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 55, 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]

(248−49, 248, 644)-Net over F₂ — Digital

Digital (199, 248, 644)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−49, 248, 12252)-Net in Base 2 — Upper bound on s

There is no (199, 248, 12253)-net in base 2, because

1 times m-reduction [i] would yield (199, 247, 12253)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 226 583152 255973 197694 028215 370310 027896 505774 789926 646892 427780 912158 657279 > 2²⁴⁷ [i]