MinT - Best Known (189−57, 189, s)-Nets in Base 2

Best Known (189−57, 189, s)-Nets in Base 2

(189−57, 189, 112)-Net over F₂ — Constructive and digital

Digital (132, 189, 112)-net over F₂, using

9 times m-reduction [i] based on digital (132, 198, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 99, 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]

(189−57, 189, 168)-Net over F₂ — Digital

Digital (132, 189, 168)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−57, 189, 1145)-Net in Base 2 — Upper bound on s

There is no (132, 189, 1146)-net in base 2, because

1 times m-reduction [i] would yield (132, 188, 1146)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 397 129761 815997 881369 441501 306250 965257 244289 705541 285016 > 2¹⁸⁸ [i]