MinT - Best Known (249−81, 249, s)-Nets in Base 2

Best Known (249−81, 249, s)-Nets in Base 2

(249−81, 249, 112)-Net over F₂ — Constructive and digital

Digital (168, 249, 112)-net over F₂, using

t-expansion [i] based on digital (163, 249, 112)-net over F₂, using
- 11 times m-reduction [i] based on digital (163, 260, 112)-net over F₂, using
  - trace code for nets [i] based on digital (33, 130, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(249−81, 249, 178)-Net over F₂ — Digital

Digital (168, 249, 178)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−81, 249, 1101)-Net in Base 2 — Upper bound on s

There is no (168, 249, 1102)-net in base 2, because

1 times m-reduction [i] would yield (168, 248, 1102)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 466 409602 934679 848380 615277 805557 587994 822391 379435 216105 617233 319228 299538 > 2²⁴⁸ [i]