MinT - Best Known (166−47, 166, s)-Nets in Base 2

Best Known (166−47, 166, s)-Nets in Base 2

(166−47, 166, 112)-Net over F₂ — Constructive and digital

Digital (119, 166, 112)-net over F₂, using

6 times m-reduction [i] based on digital (119, 172, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 86, 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]

(166−47, 166, 175)-Net over F₂ — Digital

Digital (119, 166, 175)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−47, 166, 1327)-Net in Base 2 — Upper bound on s

There is no (119, 166, 1328)-net in base 2, because

1 times m-reduction [i] would yield (119, 165, 1328)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 46 857140 662856 147008 513689 081676 681543 323245 633538 > 2¹⁶⁵ [i]