MinT - Best Known (248−109, 248, s)-Nets in Base 2

Best Known (248−109, 248, s)-Nets in Base 2

(248−109, 248, 67)-Net over F₂ — Constructive and digital

Digital (139, 248, 67)-net over F₂, using

1 times m-reduction [i] based on digital (139, 249, 67)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 94, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 155, 34)-net over F₂, using
    - net from sequence [i] based on digital (45, 33)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 1 place with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]

(248−109, 248, 89)-Net over F₂ — Digital

Digital (139, 248, 89)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−109, 248, 424)-Net in Base 2 — Upper bound on s

There is no (139, 248, 425)-net in base 2, because

1 times m-reduction [i] would yield (139, 247, 425)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 241 663701 014543 547247 249984 368178 066430 247100 891257 554448 231015 400764 760288 > 2²⁴⁷ [i]