MinT - Best Known (177−86, 177, s)-Nets in Base 2

Best Known (177−86, 177, s)-Nets in Base 2

Digital (91, 177, 53)-net over F₂, using

t-expansion [i] based on digital (90, 177, 53)-net over F₂, using
- net from sequence [i] based on digital (90, 52)-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 4 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 (91, 177, 57)-net over F₂, using

t-expansion [i] based on digital (83, 177, 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 (91, 177, 205)-net over F₂, because

2 times m-reduction [i] would yield digital (91, 175, 205)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷⁵, 205, F₂, 84) (dual of [205, 30, 85]-code), but
  - 4 times code embedding in larger space [i] would yield linear OA(2¹⁷⁹, 209, F₂, 84) (dual of [209, 30, 85]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹⁸⁰, 210, F₂, 85) (dual of [210, 30, 86]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

There is no (91, 177, 235)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 205355 843704 710460 170754 562531 290056 913277 301109 340930 > 2¹⁷⁷ [i]