MinT - Best Known (199−98, 199, s)-Nets in Base 2

Best Known (199−98, 199, s)-Nets in Base 2

Digital (101, 199, 55)-net over F₂, using

t-expansion [i] based on digital (100, 199, 55)-net over F₂, using
- net from sequence [i] based on digital (100, 54)-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 6 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 (101, 199, 65)-net over F₂, using

t-expansion [i] based on digital (95, 199, 65)-net over F₂, using
- net from sequence [i] based on digital (95, 64)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 95 and N(F) ≥ 65, using
    - Drinfeld modules of rank 1 [i]

There is no digital (101, 199, 219)-net over F₂, because

6 times m-reduction [i] would yield digital (101, 193, 219)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁹³, 219, F₂, 92) (dual of [219, 26, 93]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2¹⁹⁴, 220, F₂, 92) (dual of [220, 26, 93]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹⁹⁵, 221, F₂, 93) (dual of [221, 26, 94]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

There is no (101, 199, 253)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 822568 715573 844357 875069 309921 972777 664675 949154 468575 445518 > 2¹⁹⁹ [i]