MinT - Best Known (219−72, 219, s)-Nets in Base 2

Best Known (219−72, 219, s)-Nets in Base 2

(219−72, 219, 112)-Net over F₂ — Constructive and digital

Digital (147, 219, 112)-net over F₂, using

9 times m-reduction [i] based on digital (147, 228, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 114, 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]

(219−72, 219, 156)-Net over F₂ — Digital

Digital (147, 219, 156)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−72, 219, 916)-Net in Base 2 — Upper bound on s

There is no (147, 219, 917)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 871254 978907 680451 400952 937658 649730 347464 232721 138456 950784 374236 > 2²¹⁹ [i]