MinT - Best Known (183−89, 183, s)-Nets in Base 2

Best Known (183−89, 183, s)-Nets in Base 2

Digital (94, 183, 53)-net over F₂, using

t-expansion [i] based on digital (90, 183, 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 (94, 183, 60)-net over F₂, using

t-expansion [i] based on digital (92, 183, 60)-net over F₂, using
- net from sequence [i] based on digital (92, 59)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 92 and N(F) ≥ 60, using
    - Drinfeld modules of rank 1 [i]

There is no digital (94, 183, 208)-net over F₂, because

5 times m-reduction [i] would yield digital (94, 178, 208)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷⁸, 208, F₂, 84) (dual of [208, 30, 85]-code), but
  - 1 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 (94, 183, 244)-net in base 2, because

1 times m-reduction [i] would yield (94, 182, 244)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6 366833 589203 079036 720387 998822 449883 866999 721698 543876 > 2¹⁸² [i]