MinT - Best Known (89, 172, s)-Nets in Base 2

Best Known (89, 172, s)-Nets in Base 2

Digital (89, 172, 52)-net over F₂, using

t-expansion [i] based on digital (85, 172, 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 (89, 172, 57)-net over F₂, using

t-expansion [i] based on digital (83, 172, 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 (89, 172, 195)-net over F₂, because

1 times m-reduction [i] would yield digital (89, 171, 195)-net over F₂, but
- extracting embedded orthogonal array [i] would yield linear OA(2¹⁷¹, 195, F₂, 82) (dual of [195, 24, 83]-code), but
  - 1 times code embedding in larger space [i] would yield linear OA(2¹⁷², 196, F₂, 82) (dual of [196, 24, 83]-code), but
    - adding a parity check bit [i] would yield linear OA(2¹⁷³, 197, F₂, 83) (dual of [197, 24, 84]-code), but
      - â€œBKâ€ bound on codes from Brouwerâ€™s database [i]

There is no (89, 172, 235)-net in base 2, because

1 times m-reduction [i] would yield (89, 171, 235)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3017 196178 966976 584036 291913 306357 260066 929161 101340 > 2¹⁷¹ [i]