MinT - Best Known (166−79, 166, s)-Nets in Base 2

Best Known (166−79, 166, s)-Nets in Base 2

Digital (87, 166, 52)-net over F₂, using

t-expansion [i] based on digital (85, 166, 52)-net over F₂, using
- net from sequence [i] based on digital (85, 51)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 3 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

Digital (87, 166, 57)-net over F₂, using

t-expansion [i] based on digital (83, 166, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

There is no digital (87, 166, 194)-net over F₂, because

1 times m-reduction [i] would yield digital (87, 165, 194)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁶⁵, 194, F₂, 78) (dual of [194, 29, 79]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2¹⁶⁶, 195, F₂, 78) (dual of [195, 29, 79]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹⁶⁷, 196, F₂, 79) (dual of [196, 29, 80]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

There is no (87, 166, 236)-net in base 2, because

1 times m-reduction [i] would yield (87, 165, 236)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 47 575567 800628 888508 248599 992311 625100 466294 751258 > 2¹⁶⁵ [i]