MinT - Best Known (248−57, 248, s)-Nets in Base 2

Best Known (248−57, 248, s)-Nets in Base 2

(248−57, 248, 205)-Net over F₂ — Constructive and digital

Digital (191, 248, 205)-net over F₂, using

2¹ times duplication [i] based on digital (190, 247, 205)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 34, 10)-net over F₂, using
    - net from sequence [i] based on digital (6, 9)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 6 and N(F) ≥ 10, using
        an explicitly constructive algebraic function field [i]
  2. digital (156, 213, 195)-net over F₂, using
    - trace code for nets [i] based on digital (14, 71, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]

(248−57, 248, 420)-Net over F₂ — Digital

Digital (191, 248, 420)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(248−57, 248, 5069)-Net in Base 2 — Upper bound on s

There is no (191, 248, 5070)-net in base 2, because

1 times m-reduction [i] would yield (191, 247, 5070)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 226 385245 135309 265510 116332 296868 955618 131035 283347 021277 748873 507131 328284 > 2²⁴⁷ [i]