MinT - Best Known (249−57, 249, s)-Nets in Base 2

Best Known (249−57, 249, s)-Nets in Base 2

(249−57, 249, 206)-Net over F₂ — Constructive and digital

Digital (192, 249, 206)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (8, 36, 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]
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]

(249−57, 249, 426)-Net over F₂ — Digital

Digital (192, 249, 426)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−57, 249, 5197)-Net in Base 2 — Upper bound on s

There is no (192, 249, 5198)-net in base 2, because

1 times m-reduction [i] would yield (192, 248, 5198)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 452 508910 083572 450828 255224 358212 483996 201335 575984 669646 503225 448516 012132 > 2²⁴⁸ [i]