MinT - Best Known (246−77, 246, s)-Nets in Base 2

Best Known (246−77, 246, s)-Nets in Base 2

(246−77, 246, 112)-Net over F₂ — Constructive and digital

Digital (169, 246, 112)-net over F₂, using

t-expansion [i] based on digital (163, 246, 112)-net over F₂, using
- 14 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]

(246−77, 246, 193)-Net over F₂ — Digital

Digital (169, 246, 193)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(246−77, 246, 1255)-Net in Base 2 — Upper bound on s

There is no (169, 246, 1256)-net in base 2, because

1 times m-reduction [i] would yield (169, 245, 1256)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 56 956720 489761 202278 901691 401808 059572 495555 536378 060701 382864 551458 512973 > 2²⁴⁵ [i]