MinT - Best Known (249−77, 249, s)-Nets in Base 2

Best Known (249−77, 249, s)-Nets in Base 2

(249−77, 249, 112)-Net over F₂ — Constructive and digital

Digital (172, 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]

(249−77, 249, 200)-Net over F₂ — Digital

Digital (172, 249, 200)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−77, 249, 1329)-Net in Base 2 — Upper bound on s

There is no (172, 249, 1330)-net in base 2, because

1 times m-reduction [i] would yield (172, 248, 1330)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 459 341640 017590 401316 055359 510932 961708 963689 271489 499368 783164 726300 740040 > 2²⁴⁸ [i]