MinT - Best Known (219−75, 219, s)-Nets in Base 2

Best Known (219−75, 219, s)-Nets in Base 2

(219−75, 219, 112)-Net over F₂ — Constructive and digital

Digital (144, 219, 112)-net over F₂, using

3 times m-reduction [i] based on digital (144, 222, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 111, 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]

(219−75, 219, 142)-Net over F₂ — Digital

Digital (144, 219, 142)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−75, 219, 816)-Net in Base 2 — Upper bound on s

There is no (144, 219, 817)-net in base 2, because

1 times m-reduction [i] would yield (144, 218, 817)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 428557 222048 038574 224077 876920 858068 052077 705955 521753 037495 445048 > 2²¹⁸ [i]